They wanted to know because everyone knows that Midoriya was Quirkless, and so does All Might.
But why do they care about Midoriya's Quirk so much? Is it because All Might was Quirkless too when he was Midoriya's age?
As we find out in Episode 1 of the anime, most people manifest their quirk by age 4. It would be uncommon for anyone to manifest their quirk after this, and probably unheard of for a quirk to be manifested as late as high school age (15-16 years).
As a result, when people find out that Midoriya miraculously manifested a quirk at this late age, especially one as powerful as his, they are naturally interested as to what circumstances led to this happening. For most people, it's nothing specific to Midoriya, it's just the circumstances of his quirk. Of course, Bakugo is a lot more obsessed, but that's because of his long history with Midoriya combined with his superiority complex.
As pointed out by @TheGamer007 in the comments, there might actually not be a lot of people that know Midoriya was previously quirkless.
TheGamer007: ...in hindsight, the attention focused on Midoriya could be because normal quirks aren't as self-destructive as his. I recall someone saying during the Sports Festival arc that "it's like he's just getting used to his quirk".
Well everyone cares about Midoria's quirk because any naturally developed quirk would not cause that much damage to his user so it's possible that everyone has their own doubt about his quirk and in the starting episodes when Bokugo found out that Midoria used his quirk to pass the exam he tells everyone that he was quirkless when he was a kid even if they don't believe it but this two reason's are enough to trigger their curiosity.