As said by other answerers, it's typically the station announcing 'I'm ready for more contacts at this time.' It doesn't necessarily have to be a DX station, but any station that is attracting a lot of traffic during contests. It is kind of informal usage, as Kevid Reid mentioned, the official QRZ definition is 'who is calling me?', and CQ means 'calling all stations'. The contest station will usually only use CQ if they're not getting responses.
Typically, when a contest station says QRZ, you should reply by saying your callsign once, and quickly, especially if there is a pile-up. If the operator heard you, he will repeat back your callsign, and/or parts of it if he/she only caught part of it. It is considered very rude to repeat your callsign many times. One of the worst examples I've heard was after a DX station said QRZ, a station responded with their callsign and 'I know you're hearing me, just answer and I'll stop calling!' and 'I'm just going to keep calling until you answer me!'
This just causes more irritation to those trying to communicate, especially when the operator is talking over those who are in communication. So best practice is to say your callsign once. If you're having trouble getting through, patience is the best way to get the contact. Just keep trying, saying your callsign once immediately after they say 'QRZ' Don't be discouraged if you hear stronger, clearer stations first. They'll get serviced, and the station will get around to you. They often want to make the contact as much as you do, so they'll be on the air for a while until the pile-up dies down!
Feel free to say your callsign after either CQ or QRZ. The best thing to do is to listen for a while, and notice which stations are being responded to. Sometime operators will prematurely respond with their callsign, without waiting to hear if the other station has finished broadcasting. It's about %80 timing and %20 luck!
And, as always, good luck and 73!