The groups system is very glitchy and hasn't had much work done it in a long time. But it still technically functions. As a moderator I can't actually join groups, but I think it goes something like this:

Find the group on https://powdertoy.co.uk/Groups/Page/Groups.html, or better, find a direct link to the group in a forum thread.
Apply to the group. As far as I know you don't need to fill out the textboxes because they can't read it anyway.
Wait for someone to accept your registration. If you applied to a group being actively advertised, there is probably someone there to do that.