Making friends on sites like this can sometimes yield invites. There are also usually communities specifically dedicated to giving out invites. And if all else fails, some trackers have open invitations (either time-limited or with some kind of interview).

A common method is to get your feet wet with an easier to get into private tracker and use the invitation threads in it to get into the ones you want.

I was able to ask for an invite on a public forum, and I was able to use my demonoid (remember demonoid?) stats as proof back when they were semi private.

if you can show that you are a good and consistent uploader someone usually will sling an invite your way.

A lot of them have interviews in their IRC channel. That's how I got into MAM.

Some sometimes open up for a time.

Probably not a very practical answer, but I moved to a city known for being full of tech bros, then went to a bunch of general interest meetups organized mostly by tech bros, where I made friends with tech bros, and one of them randomly offered me an invite.