Rock climbing shoes are worn the same way normal shoes are worn, just without socks. Their sizing, though, is a different matter.
Rock climbing shoes are very tight in comparison to regular sport shoes, this is to provide better performance while climbing. This extra tightness is a little bit weird at first, but eventually you get used to it and the discomfort goes away.
You normally wouldn't wear socks in climbing shoes to make sure that your foot does not move around inside the shoe, which could reduce its performance.
Most climbers like to take the heel off in between routes. This is because the shoes are tight, so taking the heel out just reduces that stress on the feet and helps get you ready for the next route, and allows you to climb longer with less pain in your feet.
