I'm a furry artist, drawing mostly inflation/popping, so here's my take:
First, it has to be said that in the furry community the fetish is very prominent (which most of you probably already know) and therefore gives ample opportunities to analyse psychologically. However, I'll only be speaking about my own perception of it.
To me (as well as many other inflata-furs) the whole act of inflating, as well as popping, is strictly cartooney. The inflating character is empty inside and doesn't have any organs, unless it's necessitated by a gag, e.g.: opening ones chest up like a cupboard door to reveal a ideographic heart (not even a real one).
The popping, firstly, exists as a looming threat of what may come, if the indulgence into inflation goes to far. Kind of like pleasuring oneself when you're not supposed to, risking making a mess. Overall kinkiness.
In some scenarios it might be the aim altogether, that represents a form of sexual debauchery.
Later, popping is an obvious allegory of a climax: it's loud, spectacular and is the very conclusion of the 'fun'.
What I haven't seen mentioned here, is that often 'poppers' romanticise the popping to be an experience far more intense then just an orgasm. The feeling of euphoric pleasure is supposed to be felt all over the body, especially in the places of tearing (it also needs to be noted that it's not imagined as a tearing of skin, but rather that of stretched rubber). And even though it's obviously illogical, the character usually feels everything happening to all the now separate parts of themselves during the process.
The aspect of pain is virtually non-existent, unless a person is also into that sort of thing, which is generally rare. Normally, the lack of pain is all explained away with cartoon logi or altogether ignored.
Another thing is the sudden release of pent up pressure, that accompanies the feelings of climax - so a feeling of great relief.
Another one is the fact of making a scene that usually is embarrassing in its silliness. That one hints at some form of humiliation masochism (as if comically bloating up wasn't already doing that).
Yet another thing is the core of the explosive sensation being concentrated in the pit of the belly as a sexual core. This, in turn, hints at alvinolagnia - a fetish for bellies. In most cases, if you have an inflation fetish, you have alvinolagnia as well.
After the popping the character might be either alive, but magically snapped back to their original form, alive, but in pieces like some kind of cartoon character or simply gone, with only rags of rubber lying all over the place.
The concept of death is widely ignored, unless the person has a thing for snuff.
What's notable in that regard is that folks, who have a gore fetish, on-top of popping, don't necessarily combine it with snuff. They just like a visceral mess combined with cartoon logic of the inflatee surviving their blowing to bits.
Sometimes it's a nice addition to the fantasy when a post-popping scene is considered. It adds to the kinky humiliation, if the character, who popped you rubs it in your face or if a crowd of onlookers laughs at you. Lately I've seen some scenarios, where the poppees (holy word-creation batman!) head remains on the ground, flustered or angry, but otherwise alive and well, hinting at the character needing to be put together at a later time.
The source of this particular kink can originate in the various things, but is usually an exposure to a particular scene in some sort of media: cartoons, films, tales.
What started it for me was an old Polish legend about the dragon of Wawel Hill, wherein an evil dragon gets tricked into eating a fake sheep filled with sulphur, which makes him drink loads of water from the Vistula river until he blows up and explodes. Later exposure to various inflation scenes in cartoons cemented how it should look and feel.
The dichotomy of an indomitable creature, like a dragon, getting tricked into slaying themselves in such a stupid way is virtually a kin in itself. It makes the popping and the very act of inflation seem like it's well deserved and fitting. That adds to the humiliation, which yet again underlines the masochistic nature of this fetish.
That's the furry aspect of it. And it's a one shared by many other furries, who draw inflation.
I hope this little tl;dr adds a bit more context to what we poppers actually like about this seemingly horrifying fetish. It's actually all a goof in fantasy, spinned by a sexual drive and none of us would really want it to happen to us or anybody else in real life.