The stock cooler on the Intel® Core™ i5-3570K is fine "until" you start to overclock as long you have have good air flow in the case. The easiest way to think of the stock cooler is to think that it is built for average systems and performance. So if you are looking for a quiet system or better cooling you are going to want to get a new 3rd HSF (heatsink/fan).
Once you have installed the stock cooler to make sure that you have it seated well reach in and pull each of the twist clips straight up. If you are able to pull any of them straight up you didnt have it seated well and are going to want to try to relock it in place.