Why do we need Proof of Work in bitcoin?
Imagine I have 1 bitcoin. And imagine I can form a transaction to send that bitcoin to Alice or I can form a transaction to send that bitcoin to Charlie.

Now, what stops me from forming both transactions? Nothing.

So, if I do that, how will people know which transaction is valid? Clearly, without some reliable way to tell which of those two transactions are valid, the system is doomed.

This is what PoW does. A miner cannot contribute PoW to both transactions at the same time -- he must pick one.