Here is one explanation —
The Am consists of the tones A (1), C (3) and E (5). In a slash chord the bass can sometimes be part of the chord, but doesn’t necessarily have to. In this case the bass note D is not a part of the Am chord written on the left.
The Am/D chord can also be considered as D7sus2 chord. Or as D7(9) without the 3rd being played. Therefore you can play this chord instead of D7 (D9, D13) or D7sus (sus2, sus4, sus9, etc.). Practically it will most likely be used instead of Dsus9 (and you can normally as well apply Dsus9 where Am/D is written). Dsus9 can be interpreted as Am7/D (considering the 5th A of Dsus9 being played). Usually you’ll more likely interpret/write Dsus9 as C/D.