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.

The organization and typical submission requirements are what really put them over public trackers for me.

Public tracker: It’s this big and this many files. Figure it out.*
Private tracker: All the metadata

* Experience may vary. Post is overly dramatic for comedic effect