'Clingy' isn't an absolute sort of thing. It's relative.
Like with the above two posters, they desire a high amount of attention, and don't mind the space/time sharing. If their romantic interest has roughly the same desire for attention/etc, then they match, and there is no significant discrepancy. Clinginess therefore does not occur in anyone's view.
Though, if their fiance were particularly independent, with high self-esteem, and a lot of ways in which they want to spend their time, things might not go so well.
So imagine a guy or girl with particularly high self esteem, who derives some measure of comfort from their partner, but does not need that much. The rest of their time, they'd prefer to spend on other things. If they met someone with high attention demands, then there is a discrepancy, and clinginess is perceived.
For many people, the idea of being relied upon can get old, and eventually frustrating. Being relied upon represents a limitation to one's independence. Depending on how important independence is to a person, that reliance/limitation can create an enormous stress, a discomfort, a frustration. This is also compounded when one particularly values independence in themselves and others: The smothered person will lose respect for the other, because their lack of independence is so prominent. It's a weakness that gets perceived as supremely unattractive.
Then there's the perspective of responsibility. When a person emotionally relies upon you, you have become, without your own permission, responsible for their emotional state. The things you do will assuredly affect the others emotions profoundly. And if you put them in a bad mood or do not ameliorate a bad mood that has nothing to do with you, their reactions will generally impose upon your mood(often purposefully).
Now, this isn't so different from a normal relationship, you might say. True. But there is a big difference. In a healthy relationship, your actions will affect the mood of your partner by [X]. In a clingy relationship, your actions will affect the mood of your partner by [3X].
So, getting back to responsibility..... Who wants that responsibility?
...Well, though that's a rhetorical question, I guess the answer is: Those not in a position to be picky.
All that said, this dynamic shifts after having been involved with someone a long time. But within the world of young adults, it is a reliable paradigm.