How should the two As be played with the left hand? Why is the second A even necessary?
The two As are there to indicate that the left hand is playing two (logical) voices. One has an A half note while the other has a A quarter note followed by a G quarter note.
In keyboard terms, this means that you play an A on the third beat and, without releasing the A, a G on the fourth beat. Release both notes at the end of the measure.
If you look at the entire measure, the two voices are of course even clearer. If the lower voice had a rest on the third beat, you would play this on a keyboard in essentially the same way. But if you were transcribing this for four instruments, the result would be noticeably different.
Conversely, if this is a keyboard reduction of a piece composed for an ensemble of singers or instruments, it would be helpful to retain the information that the bass part has a quarter note there, not a rest.