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 thought my cat was strange for this same behavior. I should have known he wasn’t the only one. He will borrow under the blanket himself if I don’t have an entrance for him. Or paw at my leg until I make him one.