I usually like to watch the first few picks. Its a special moment for the players drafted and I enjoy reminiscing on those moments when I watch their careers. Plus, now that I've been watching a while. Its crazy to be able to remember the moment a player was drafted to seeing them retire.
Also, there is some added drama. Some guys fall and start their career with a chip on their shoulder. Some may have a 2nd/3rd round grade and have to prove their team made the right decision. Teams may trade up to land their guy and make a statement about their conviction on that player. Some teams get fleeced. I always like to hear about some of the inspiring background stories of some of the players as well.
All in all, I think it adds to the experience of being an NFL football fan. You become familiar with some of the players entering the league, cheer some of them on, and connect with some of the players even if you aren't a fan of their specific team.
Plus, I like to laugh at the Packer's picks...until last year, they were on a run of terrible 1st round picks.