After watching every Don't Hug Me I'm Scared episode about 20 times (that's not an exaggeration; I'm slowly becoming obsessed with this show), I can conclude it has two meanings.
The first is what it means in our world. It's obvious that it's about the media and how different age groups (Yellow being a kid, Red being a teen and Duck being an elder) are affected by it. The singing puppets represent the media and it's hypocrisy, and while Yellow Guy is entranced by it, as kids can be with toy commercials and stuff, Red and Duck are doubtful, as people do become when they get older. Red escaping and then unplugging the machine represents him making the decision to not let the media control him, and to live his life as he pleases. However, that ending scene where everything resets and the notepad starts singing again implies that there is no escape from it, and commercialism is there with you where you go.
However, I believe there is a second meaning, and that's about what is literally happening, if that's possible. This is about what is actually happening to the puppets and their story, and the significance behind things such as the date of June 19th and Roy. In a rather short theory, I believe that Roy kidnapped the three puppets (on Father's Day, which is June 19th in England, the country of origin), has trapped them in a TV studio (based on the fact that we see the kitchen has no ceiling in the 5th) and is forcing them to film a TV show with corrupted lessons and morals. And when the puppets do not cooperate, Roy punishes them, explaining the warped and creepy endings every episode has that usually ends in the puppets being killed or tormented. For example, Tony screeching and making Yellow Guy’s ears bleed when the three question his lessons, followed by him speeding up time and killing them, Duck being slaughtered and made into food for trying to escape, Colin trapping the three in the Digital World after Red Guy defies him and more. Also, Roy making small cameos in every episode also supports the theory, as it's his way of watching them and reminding them that if they do not keep in line and do what he asks, they will be punished. We know the puppets have been kidnapped because in the "HELP" kickstarter videos, we have seen them tied up and being held captive by Money Man, who is most likely an associate of Roy and the one who brought the three puppets to him. In fact, in one of the promo videos, Yellow Guy even asks "why are we at my dad's house?" Roy uses the machine seen at the end of 6 to create the teachers, also, but that's pretty obvious, as we have seen. Now, why is Roy doing this? I believe it is to bully or abuse his son, Yellow Guy, whom he is not proud of and gets joy out of torturing. In fact, in the DHMIS interview on the site "It's Nice That," Roy even says the line "My silly boy has allowed his eyes to grow arrogant and rude, for this I will take him on a trip to punish land." This supports why the puppets seem to have an out for him in particular, with Tony making his ears bleed, Shringold singling him out and trying to get him to join the cult and the sketchbook destroying his painting of the clown. Also, both Red and Duck act like they are in a TV show, with pretty bad acting to boot. For example, the two clearly trying to cue the globe’s song at the beginning of the fourth one, Red Guy looking right at the screen and/or saying some things slowly with strange gestures, Duck staring at the screen for a few seconds after his “line” in the Love Song, and more. I believe it is because they are unaware of what to do and thus act along so they don't get hurt, but they are terrified so they act poorly. Now, my take on the final scenes of the last episode is interesting. I believe that the scenes with Red Guy working in an office and singing at the bar are flashbacks to when Red Guy was kidnapped and had his first encounter with Roy. He seems to have ideas for a children's show, pitching the ideas of inanimate objects, such as the file, singing educational songs. Roy was at the bar when Red Guy pitched his idea for the Creativity Song, and he kidnapped him that night and then used that idea to create his torture device for the three. And when Red Guy is in the control room, that is him discovering Roy's headquarters after escaping at the end of the 4th, and when he unplugs it he is saving Yellow Guy and ending their terror once and for all. And the final scene with the different-colored puppets? I think that could be a way of saying that Roy kidnapped new victims who bear resemblance to the last three, and they will undergo the torture that happened in the last six episodes. As for Duck? He is dead, and Roy did indeed kill him at the end of 5 for trying to escape. However, the other two got away, and now they are free from Roy's sick grip.
EDIT: And one tiny other idea which I admit I got from Game Theory - Roy's profession is in the food business, specifically grain-based foods. That's why things such as boxes of oats, flakes and bread often appear in the series, and the Health Band from episode 5 seems to list only grain-based foods as the healthy ones. Roy is using product placement in his show to advertise his brand. So this show is not only a way of torturing his captives, but also advertising his business.
*Panting* Aaaaaaaand that is my theory on the show. If you read all this, you are an amazing person and I thank you so much. Feel free to leave your thoughts below.
And thanks for reading!