Bitcoin Forum

I've heard somewhere that leaving the Bitcoin Core app running contributes to the network? If so how does it work? What does it do?

You can inform yourself about this on the Bitcoin Wiki, there a page on why your should run a node (https://en.bitcoin.it/wiki/Full_node#Why_should_you_run_a_full_node.3F).

Bitcoin Core is the backbone of the network. A full node enforces the rules of the network and helps to decentralise the network. A full node will not trust one another and they have a set of rules to follow. They will check each block for the validity and whether it follows the rules the client enforces. Without them checking the rules, a miner can easily change the protocol rules such as the maximum market cap. Just running Bitcoin Core will do this, no need to port forward.

Yes, that is true.

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.