What does it mean to dampen the inspiration?

Inspiration is the moment of creativity, the birth of a new idea. To dampen is to make a little wet, as from the rain.

So when you dampen someone's inspiration, you're holding them back from taking full advantage of their great idea. Usually, this is by telling them something negative ("It's not so new"), realistic ("It's been done before") or both ("It won't work").

Sometimes dampening inspiration is a good thing, as it shows someone the flaws with their plan or idea before they get too involved with it. But most of the time, it's a bad idea to stifle someone's creativity: it leads to hurt feelings, and might even hold them back from being inventive (or from sharing their ideas) in the future.