The key being that your index finger covers every string on the first fret (i.e., it's a barre chord). You'll notice that the rest of the chord looks just like a Gm, shape-wise, which makes sense when you think about it. You're basically just playing a Gm one fret higher.
One super-easy way you can play a G#m is X876. You're using the same shape as an Em chord, except four frets higher. Don't play the fourth string (G string) at all. The downside is that if the other chords you're playing are further down the neck, that version of G#m will tend to "stick out" too much, due to how high the notes are arranged.
The trouble with songs that contain G#m chords is that they usually contain other tricky chords too. Probably the most (in)famous song right now that features a G#m is "I'm Yours", assuming you play it in the original key of B: