Across the map of Popgoes Arcade, there are numerous gift boxes that only appear and can only be seen when you are wearing their associated color, with one associated with each paint bucket that can be found and worn. The gift boxes ONLY appear if you have beaten the Puppet and are wearing their ribbon (Thanks @Rojax!). Gift boxes are spread throughout the entire map of Popgoes Arcade, bar one specific location.

The gift boxes will look like the attached gift box to the right, which is associated with the Violet paint, with the ribbons taking the color of their associated paint. Each gift box provides dialogue with a similar structure but different contents and subjects.

Gift boxes 1-5 can be found in Popgoes: The Dead Forest or Popgoes and the Machinist. Gift boxes are not locked out from being found when progressing from Dead Forest to Machinist, I believe.

The coordinate map of Popgoes Arcade, for ease of describing where each gift box is.

Located in the South Forest at coordinates (2,2).
Upon opening the box, the following dialogue appears:

1. "THE BOX IS OPEN.
   
2. A CALMING, FRESH, GREEN GLOW SHINES FROM INSIDE IT.
   
3. IT SMELLS LIKE A GARDEN. LIKE CUT GRASS. LIKE APPLES.
   
4. THOUGH NOTHING TANGIBLE IS HOUSED IN THE BOX, IT GIVES OFF A SENSATION.
   
5. THE BOX FEELS... FRIENDLY.
   
6. IT IS COMFORTING AND REASSURING. A SENSE OF CARING FOR FRIENDS, FAMILY, AND THEIR JOURNEYS.
   
7. BUT, AT THE SAME TIME...
   
8. A REMINDER OF CRUSHING RESPONSIBILITY.
   
9. ANXIETY. FEAR OF NOT MEETING EXPECTATIONS. TO BE DEPENDABLE. TO BE A HERO. THE HERO."

The bold, underlined letters glitch in pairs in the dialogue box. The letters glitch in the following pairs:

• I (from INSIDE, Line 2) & T (from RESPONSIBILITY, Line 8)
  
• I (from LIKE, Line 3) & F (from OF, Line 8)
  
• I (from NOTHING, Line 4) & S (from RESPONSIBILITY, Line 8)

Located in the North Forest at coordinates (9,10).
Upon opening the box, the following dialogue appears:

1. "THE BOX IS OPEN.
   
2. A SHARP, NEON, YELLOW GLOW FLICKERS FROM INSIDE IT.
   
3. IT SMELLS LIKE THE AIR AFTER THUNDER.
   
4. THOUGH NOTHING ACTUALLY SITS IN THE BOX, IT TRIGGERS AN UNDENIABLE SENSATION.
   
5. THE BOX FEELS... INTELLIGENT.
   
6. A SENSE OF PURPOSE, IMPRESSIVE KNOWLEDGE AND SKILL, AND PUT TO GOOD USE.
   
7. BUT, AT THE SAME TIME...
   
8. A CONCERN OF BEING NOTHING MORE.
   
9. BEING USEFUL, OR BEING USED? BEING RESOURCEFUL, OR BEING A RESOURCE?"

The bold, underlined letters glitch in pairs in the dialogue box. The letters glitch in the following pairs:

• H (from THUNDER, Line 3) & I (from BEING, Line 9)
  
• I (from AIR, Line 3) & R (from PURPOSE, Line 6)
  
• E (from TRIGGERS, Line 4) & C (from CONCERN, Line 8)

Located in the East Forest at coordinates (9,4).
Upon opening the box, the following dialogue appears:

1. "THE BOX IS OPEN.
   
2. A SMOOTH, PULSATING, BLUE GLOW FLOWS FROM INSIDE IT.
   
3. IT SMELLS LIKE LIKE INK. PICTURES AND PASSAGES IN BOOKS AND LETTERS.
   
4. THOUGH THE BOX CLEARLY HAS NO PHYSICAL CONTENTS, A VIVID SENSATION EMANATES FROM IT.
   
5. THE BOX FEELS... TRUSTWORTHY.
   
6. RELIABLE. LEVEL-HEADED, HONEST. CONFIDENT LEADERSHIP, CLEAR COMMUNICATION.
   
7. BUT, AT THE SAME TIME...
   
8. DESPERATION TO NOT BE BORING.
   
9. TO NOT BE FORGETTABLE, BLAND, OR A BUZZKILL. TO BE AS EXCITING, AND FUN, AS EVERYONE ELSE"

The bold, underlined letters glitch in pairs in the dialogue box. The letters glitch in the following pairs:

• R (from PICTURES, Line 3) & D (from BLAND, Line 9)
  
• S (from PASSAGES, Line 3) & T (from DESPERATION, Line 8)
  
• O (from FROM, Line 2) & N (from CONFIDENT, Line 6)

Located in the North Forest at coordinates (3,11).
Upon opening the box, the following dialogue appears:

1. "THE BOX IS OPEN.
   
2. A HARSH, FLUORESCENT, PINK GLOW FLASHES FROM INSIDE IT.
   
3. IT SMELLS LIKE SWEETS, CANDY AND CAKES AND FIZZY DRINKS
   
4. THOUGH THE BOX IS EVIDENTLY EMPTY, A SENSATION SPARKS FROM IT AFTER BEING OPENED.
   
5. THE BOX FEELS... CREATIVE.
   
6. A TRUE, LOUD, PASSION FOR ART. FOR ENTERTAINMENT, FOR FUN, AND TO SEND A MESSAGE.
   
7. BUT, AT THE SAME TIME...
   
8. A FEAR OF BEING ANNOYING.
   
9. TEETERING ON THE LINE BETWEEN CHARISMA, AND EGO. TO BE NOTICED, BUT NOT OVERBEARING."

The bold, underlined letters glitch in pairs in the dialogue box. The letters glitch in the following pairs:

• D (from CANDY, Line 3) & L (from LINE, Line 9)
  
• L (from FLUORESCENT, Line 2) & Y (from ANNOYING, Line 8)
  
• L (from GLOW, Line 2) & I (from ANNOYING, Line 8)

Located in the East Forest at coordinates (8,8).
Upon opening the box, the following dialogue appears:

1. "THE BOX IS OPEN.
   
2. A WARM, CRACKLING, RED GLOW RADIATES FROM INSIDE IT FROM INSIDE IT.
   
3. IT SMELLS LIKE A CANDLE AFTER BEING BLOWN OUT.
   
4. THOUGH THERE'S NOTHING TO TAKE OUT OF THE BOX, OPENING IT EVOKES A UNIQUE SENSATION.
   
5. THE BOX FEELS... TALENTED.
   
6. IMPRESSIVE, NICHE CAPABILITIES, AN ACTUAL ABILITY TO APPLY UNIQUE SKILLS. MANY OF THEM
   
7. BUT, AT THE SAME TIME...
   
8. A DRAINING SENSE OF UNFULFILLMENT.
   
9. SWEPT ASIDE, WITH LITTLE OPPORTUNITY TO DEMONSTRATE EXPERTISE, OR EVEN COMPETENCE."

The bold, underlined letters glitch in pairs in the dialogue box. The letters glitch in the following pairs:

• F (from FROM, Line 2) & T (from UNFULFILLMENT, Line 8)
  
• Y (from ABILITY, Line 6) & G (from DRAINING, Line 8)
  
• G (from NOTHING, Line 4) & I (from UNFULFILLMENT, Line 8)

Located in the South Forest at coordinates (6,1).
Upon opening the box, the following dialogue appears:

1. "THE BOX IS OPEN.
   
2. A SUNNY, SATURATED, ORANGE GLOW SPARKLES FROM INSIDE IT.
   
3. IT SMELLS LIKE A BOWL OF FRUIT, LIKE ORANGES, PEACHES, AND MANGOES
   
4. THOUGH THE BOX HAS NOTHING TO GRAB, ITS WALLS AND FLOOR PROJECT SOME KIND OF SENSATION.
   
5. THE BOX FEELS... PASSIONATE.
   
6. AFFECTION, RESPECT, APPRECIATION, INFATUATION: LOVE, IN ITS PUREST FORM.
   
7. BUT, AT THE SAME TIME...
   
8. A NAGGING WORRY OF REJECTION.
   
9. THE INSTANT, PAINFUL CRUSHING OF AN OBSESSIVE FIXATION THAT CANNOT BE OTHERWISE CURED."

The bold, underlined letters glitch in pairs in the dialogue box. The letters glitch in the following pairs:

• F (from FRUIT, Line 3) & T (from REJECTION, Line 8)
  
• E (from ORANGES, Line 6) & D (from CURED, Line 9)
  
• I (from PASSIONATE, Line 5) & F (from PAINFUL, Line 9)

Located in the West Forest at coordinates (1,5).
Upon opening the box, the following dialogue appears:

1. "THE BOX IS OPEN.
   
2. A BRIGHT, FROSTY, WHITE GLOW DIFFUSES FROM INSIDE IT.
   
3. IT SMELLS LIKE A DELICIOUS, HOME-COOKED, MEAL. HEARTY FOOD, MADE WITH CARE.
   
4. THOUGH THERE'S NOTHING TO TAKE FROM THE BOX, A SENSATION RADIATES FROM IT.
   
5. THE BOX FEELS... GENEROUS.
   
6. A DESIRE TO HELP, GIVE, AND PROVIDE. TO CONTRIBUTE, TO FEED, TO SHELTER. TO SIMPLY BE THERE.
   
7. BUT AT THE SAME TIME...
   
8. NOT NECESSARILY FITTING IN.
   
9. IS IT TRUE FRIENDSHIP? OR IS IT FEIGNED, FOR A REWARD? IS IT CARE, OR A TRANSACTION?"

The bold, underlined letters glitch in pairs in the dialogue box. The letters glitch in the following pairs:

• G (from GLOW, Line 2) & O (from NOT, Line 8)
  
• T (from HEARTY, Line 3) & E (from CONTRIBUTE, Line 6)
  
• E (from GENEROUS, Line 5) & D (from FEIGNED, Line 9)

Located in the West Forest , underground along the main path.
Upon opening the box, the following dialogue appears:

1. "THE BOX IS OPEN.
   
2. A SPARKLING, SPECTRAL LIGHT GLIMMERS FROM INSIDE IT.
   
3. IT SMELLS CLEAN. LIKE FRESHLY WASHED CLOTHES, HAND SOAP, AND NOTHING ELSE.
   
4. THOUGH THERE'S NOTHING PHYSICALLY IN THE BOX, IT EMITS A SENSATION.
   
5. THE BOX FEELS... LOYAL.
   
6. A CONSTANT, SELFLESS DEDICATION TO SUPPORTING ANOTHER. TO TALK, TO ASSIST. TO GENUINELY CARE.
   
7. BUT, AT THE SAME TIME...
   
8. A DEEP AND GROWING SENSE OF UNCERTAINTY.
   
9. DISCOMFORT. A CRISIS OF CONSCIENCE. SITTING, PERHAPS, ON THE WRONG SIDE. AND BEING TRAPPED THERE."

The bold, underlined letters glitch in pairs in the dialogue box. The letters glitch in the following pairs:

• L (from LIKE, Line 3) & D (from DEDICATION, Line 6)
  
• D (from INSIDE, Line 2) & A (from TALK, Line 6)
  
• A (from SPECTRAL, Line 2) & S (from SENSE, Line 8)

Located in the South Forest, underground along the main path.
Upon opening the box, the following dialogue appears:

1. "THE BOX IS OPEN.
   
2. A DEEP, ABYSSAL, INDIGO LIGHT GLISTENS FROM INSIDE IT.
   
3. IT SMELLS LIKE THE OCEAN. LIKE FISH AND SEAWEED. AND LIKE SUNSCREEN AND TOWELS DRYING IN THE SUN.
   
4. THOUGH NOTHING HAS BEEN PACKAGED IN THE BOX, A SENSATION FROM IT CAN BE FELT.
   
5. THE BOX FEELS... CURIOUS.
   
6. A DESIRE TO LEARN, TO EXPLORE, AND TO INVESTIGATE. TO BE AT THE CORE OF A MYSTERY, AND TO FIGURE IT OUT.
   
7. BUT, AT THE SAME TIME...
   
8. BEING LOST.
   
9. EXPLORING A NEW ENVIRONMENT, OR NEW INFORMATION, AND NOT KNOWING WHAT TO DO WITH IT. AT ALL."

The bold, underlined letters glitch in pairs in the dialogue box. The letters glitch in the following pairs:

• S (from GLISTENS, Line 2) & H (from WHAT, Line 9)
  
• P (from PACKAGED, Line 4) & E (from BEING, Line 8)
  
• H (from FISH, Line 3) & A (from AT, Line 6)

Located in the East Forest, underground along the main path.
Upon opening the box, the following dialogue appears:

1. "THE BOX IS OPEN.
   
2. A WARM, BUT COOL, VIOLET LIGHT BLEEDS FROM INSIDE IT.
   
3. IT SMELLS, NOT LIKE ANYTHING OF A PURPLE HUE, BUT OF METAL. OF MACHINES, TOOLS, COINS, AND RUST.
   
4. THOUGH NOTHING IN THE BOX CAN BE REACHED, IT GIVES A DISTINCT SENSATION.
   
5. THE BOX FEELS... AMBITIOUS.
   
6. AN APPETITE FOR GROWTH. FOR ADAPTING, LEARNING, EVOLVING. FOR BEING BIGGER, STRONGER, BETTER.
   
7. BUT, AT THE SAME TIME...
   
8. AN INDESCRIBABLE LONELINESS.
   
9. AWARENESS OF TOTAL, PERHAPS SUDDEN, ABANDONMENT. AND, AS A RESULT, ANGER."

The bold, underlined letters glitch in pairs in the dialogue box. The letters glitch in the following pairs:

• A (from METAL, Line 3) & R (from STRONGER, Line 6)
  
• I (from MACHINES, Line 3) & R (from GROWTH, Line 6)
  
• I (from IN, Line 4) & P (from PERHAPS, Line 9)

I do not know if the glitchy letters matter at all, whether it is part of a larger puzzle or just a bit of the visual design. One note about how the glitchy letters appear to function: Each box appears to have 3 different pairs of glitchy letters, which cycle which pair is visually glitching each time you read their associated box. Only one pair glitches at a time, per read of the box, with each read of the same box cycling through which pair glitches.

Steam User 'Michael Afton 4 DBD' proposed the following answer to the mystery of the odd letters, found by lining up each of the letters in their sequences in specific orders, the following sentences can be constructed:

I FIRSTLY GIFTED GOLD HAIR
I SECONDLY GIFTED A SHIP
I THIRDLY GIFTED A SPEAR