The notes of Bm6 are B, D, F# and G#. B is the root, D the minor third, F# the fifth and G# the sixth. Said aloud, its notes are B, D, F sharp and G sharp.
From the low string to the high one you hear B, F#, B, D and G#. Of the five strings you play, two give B, the note that names the chord. The lowest note is B on string 5 and the highest is G# on string 1.
You play Bm6 with a barre at fret 2. String 6 is not played. Everything fits between frets 2 and 4. The second shape on this page starts at fret 7.
Bm is one change away. String 1 moves from fret 4 to fret 2, bringing in F#, the fifth of Bm. It shares two notes with D, D and F#. The change that keeps the most fingers still is to Em. One stays in place.
Bm6 belongs to two keys. In A major it is the ii6 and in F# minor the iv6.