I've heard somewhere that leaving the Bitcoin Core app running contributes to the network?
Yes, that is true.If so how does it work? What does it do?
Bitcoin Core (and its derivatives) connect to other computers also running Bitcoin Core. Those computers are known as nodes. Each node, when it receives a block or transaction, verifies it fully. In order to do full trustless verification, it needs to have verified the entire blockchain. Doing that requires downloading and verifying the entire blockchain. Once a block or transaction is considered valid, the node will then relay the block or transaction to all of the nodes that it is connected to. 
