Seeing the games they review/collect/comment on?
Not missing a new release from them when they publish it?
Some other reason?

For me, it's their devlogs. I find people's progress on their games to be quite interesting.

Like Bynary Fission said above, getting to see people's devlogs is a big reason for me.

But I also follow people as a way to say "Hey I like your stuff", and so that I see what they're doing/making/following/etc. in my feed.

I mainly following folks as a way of supporting them at first. After that, I mainly care about the email I get whenever they release something new.

I follow someone here just to see if they release a new game.

I sometimes follow based on a game they made and how communicative they are about it.

When following a dev it's usually because A. I like their work. B. I want to see the progress on a project that interests me or C. I want to support them in a small way and keep updated on what they are up to.

The games they have on offer or their assets

To make indie devs feel better about their work. 

Oh, and I guess I'm also a huge fan of their work but that's a coincidence. 

I'm always a sucker for pretty pages lmao. I want to improve mine, and I think pages with great visual designs just draw me in.