I have a Shimano CS-5600 11-23 cassette....
I have a Shimano CS-5600 11-23 cassette. I want to go up to 11-25 to help with climbs. Do I need to stick with the CS-5600, or can I use any Shimano cassette?
No problem, this CS-6700 cassette and any (except 7800) Shimano 10 speed cassette and even SRAM 10 speed cassettes will work great.
Don't buy a Shimano Dura Ace 7800 cassette because it won't fit your wheels, it has an entirely worthless Dura-ace proprietary hub design.
He is correct in that The 6700 and 7900 Cassettes are both compatible with your group and both made in 11-25 combinations.
The 7800 will also work with your hub or any other Shimano 8, 9 or 10 speed hub out there. The compatibility he is referring to was not in the cassette it was a 10spd specific Freehub that was used in DA and Ultegra hubs and Wheels for a few years.
NOT the Cassettes they are compatible.