There are two key motivations for the move to Proof of Stake:

Ethereum developers and researchers believe that consensus algorithms based on Proof of Stake (PoS) can provide a higher degree of security for a given amount of resource expenditure, compared with Proof of Work (PoW) consensus algorithms. Ethereum developers and researchers believe that a new design called Casper resolves the important remaining issues with the current "state of the art" in Proof of Stake design. These issues include imperfect decentralisation and vulnerability to certain types of attacks, as well as other economic and performance considerations.

Both topics touch on issues worth elaborating in their own questions and answers, so I'll be linking to other questions as I summarise the main reasons for these two motivations.

disclaimer: I've contributed to the Casper research effort in a few small, oblique ways. However I have no formal relationship with the Ethereum project and I think my answer here is fairly objective in any case. Suggested changes are welcome.

Why would Proof of Stake provide more security for a given investment of resources than Proof of Work?

First, some background is useful. The goal of a consensus algorithm in a public blockchain network is to let many different users agree on the current state of the blockchain even though they don't trust each other or any central authority. This is a challenging problem, and until the Bitcoin network first solved it in 2008 by using Proof of Work (a.k.a. "mining", a.k.a. "hashing") no really good solutions were known. So the "performance" of the algorithm wasn't originally that important--the fact that it could be done at all was impressive enough.

After Bitcoin showed that a solution to public consensus was possible, new ideas such as "Proof of Stake" began to pop up in its wake. For a full explanation of the differences between PoW and PoS consensus, see here. Once new algorithms were proposed, the question turned from "does it work?" to "does A work better than B?"

There are several different ways to answer this question. One way to gain an insight into the potential promise of PoS protocols is to consider four factors in particular:

The cost to cooperating "honest nodes" in the absence of hostile attack The cost to cooperating "honest nodes" during a hostile attack The cost to attacking "hostile nodes" if their attack is unsuccessful The cost to attacking "hostile nodes" if their attack is successful

By comparing these costs to block rewards and other potential gains we can estimate the amount of "honest node" resources required to both run the network and hold off an attacker of a given scale. This provides a reasonably fair way to compare the level of security provided by two different consensus protocols.