D/F# Means play a D Chord with an F# bass. The F# is actually the third note in the D chord so you are playing the D chord but instead of the D being the root (or bass) note you are making the D# the root.

It is the same on the piano. With the right hand you are playing a full D chord (three notes D, F# and A) and with the left hand you are playing the F# as a bass note. If you are playing bass guitar, you simply play the F# note.