「飲ませます」 is the polite causative form of 飲む (to drink). It means to make someone drink something (e.g., medicine to a child) or to allow/let someone drink (e.g., water to a pet). Example: 子供に薬を飲ませます (Kodomo ni kusuri o nomasemasu - I will make my child drink the medicine).

「飲む」の丁寧な使役形です。誰かに何かを飲ませる（強制や許可）という意味になります。