Collators maintain parachains by collecting parachain transactions from users and producing state transition proofs for Relay Chain validators. In other words, collators maintain parachains by aggregating parachain transactions into parachain block candidates and producing state transition proofs for validators based on those blocks.
A collator cannot continue building blocks on a parachain until the block candidate they proposed to the Relay Chain validators has been validated.
Collators maintain a full-node for the Relay Chain, and a full-node for their particular parachain; meaning they retain all necessary information to be able to author new blocks and execute transactions in much the same way as miners do on current PoW blockchains. Under normal circumstances, they will collate and execute transactions to create an unsealed block and provide it, together with a proof of state transition, to one or more validators responsible for proposing a parachain block.
Collators are a key element of XCMP (Cross-Chain Message Passing). By being full-nodes of the Relay Chain, they are all aware of each other as peers. This makes it possible for them to send messages from parachain A to parachain B.