So it’s basically a combination of everything that everyone else has said.

1. NFL is by far the most popular sport in the country, but it also only has 17 regular season games (versus 82ish for NHL and NBA, 40ish for MLS, and 162 for MLB), and the entire season is spread over only 6-7 months of the calendar year, versus 8-10 for the others. There is an appetite for any content at all that materially affects the most watched sport.
2. College Football itself is probably the fourth most popular sports “league” in the country, though its organization and economics are WAY different (for now) than the normal pro leagues’. There’s huge overlap in general of course, but the Draft brings all of the fans together as CFB fans see where the top players will move.
3. Going back to number 1, the NFL and media companies, being what they are, noticed the gap in the sporting calendar (after March Madness, before NBA and NHL playoffs, very early in the MLB season, MLS well… (LOL, I love MLS and it’s a miracle it’s stable but it’s still not an important “TV sport” in this country). They also noticed that a certain segment of die-hards have been watching the draft for 30 years, and they saw an opportunity to tap that dormant interest for months of “segments” and a big day of ratings and revenue, so of course they did.
4. More recently, seeing that their hype efforts were working, they’ve moved it out of an auditorium near League HQ and made it a travelling road show, goosing local attention and furthering the image that it’s an event.

As to why all that worked, I like the posts that talk about the optimism and renewal that the draft represents. The NFL is unique in how it handles player development, in that it mostly doesn’t because it has an independently-popular lower league that will do it for free. Since that lower league is effectively the sole source of players, and since the NFL is an American-style sporting cartel, the Draft becomes the single biggest infusion of talent that a team will see in a given year, some of it ready to contribute on the field right away, and the teams that need the talent the most usually have the best picks and therefore a real chance to improve quickly, though the same bad management that gets teams in a bad place will often squander that chance.

For those who follow European football (soccer, not the niche gridiron leagues over there), imagine a single day (okay, three days now, but Rounds 2-7 are still for the nerds) that combines the anxiety of a promotion playoff final (though with deferred results) with the excitement of the summer transfer window (let’s consider NFL free agency the equivalent of the winter window).