Why do we need Proof of Work in bitcoin? I know that proof of work shows that the person has put in some time and power into the processing. I also know that bitcoin mining is adding a transaction into the blockchain and then the miner will need to solve a pow problem.

Why do we need this? What if we eliminated the proof of work step for the miners and what are the consequences of doing so?
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.