Nodes validate transactions and blocks as per the Bitcoin protocol, as a result valid unconfirmed transactions and blocks (of confirmed transactions) are relayed to neighboring nodes (peers).
So why does Bitcoin need the proof-of-work?
This is oversimplifying it, but in my mind the first reason that stands out is "minting". Minting is the issuance of new coins into the system, the proof of work is one way of controlling the inflation rate and initial distribution of coin supply.
The second reason is security or "immutability" of the blockchain. By requiring proof of work it means that if someone is attempting to cheat the Bitcoin network then they have to expend slightly more energy than everyone else who is playing by the rules, and you have to do this continuously or risk being overtaken by the "good guys" who will reject your chain of work.
In the long run the theory is that it's more rewarding to play nice than try and cheat the system, and proof of work is an essential component in making that happen.
I recommend listening to a few of Andreas Antonopoulos talks on youtube, he's very knowledgeable and verbose in his talks about the big picture and game theory behind Bitcoin.
One video that I think is along the lines of your question is:
Bitcoin Q&A: Proof-of-Work (PoW), Proof-of-Stake (PoS), Delegated Proof-of-Stake (DPoS)
https://www.youtube.com/watch?v=3W_3AQrQEOM