How does generating random numbers "remove entropy from your system"? In a previous question, I quoted the --gen-random entry in the GPG Man Pages. At the end it says:


  PLEASE, don’t use this command unless you know what you are doing; it may remove precious entropy from the system!


How does that work? I understand entropy to be the "randomness" of a system (i.e. the amount of information it contains).  How can generating random numbers steal randomness from your computer such that it noticeably hurts the computer's ability to perform cryptologic operations in the future?

I'm no mathematician, so plain english answers would be appreciated.
As others have stated, the problem is the entropy pool. The operating system maintains a count of entropy in the pool that it decreases every time random numbers are generated from it and increases when it adds some timing or other information that is assumed to have entropy.

Requesting numbers does not actually remove randomness from the pool in practice, unless it is severely broken. The state will still be unknown and the numbers received cannot be used to derive future numbers.

However, the /dev/random interface on Linux blocks if the count is zero. This means that requesting a lot of random numbers from the OS may starve any programs that use the blocking interface. This is the practical issue with using the command, since especially on server machines that have no input devices the entropy count can be slow to increase.

The usual advice is to use the non-blocking pool in most applications so that they cannot get starved, but not every program out there does that.