For centuries, the better supplied side held the advantage on the field of battle. In Acceptable Losses, it is no different.
Supplies affect your platoon every day. Low supply levels make offensive missions more difficult and can negatively impact your men during the Night Phase.
Assigning Squads to Supply Patrols is the primary means of gaining supplies. Supplies may also be found on other missions.
Every day consumes supplies. Your men must eat, and missions consume ammunition.
If your platoon runs completely out of supplies, HQ will authorize emergency reserve stores to prevent total collapse.
Bottom Line
Supplies feed your men. Supplies keep them fighting. If supply levels fall too low, every part of the campaign becomes harder.