What it means:

that's the problem

How Shakespeare used it:

In Hamlet's famous "To be or not to be" soliloquy, "ay, there's the rub" is the tormented prince's acknowledgement that death may not end his difficulties because the dead may perhaps still be troubled by dreams. (Hamlet, Act 3, Scene 1)

(The original rub predates Shakespeare. On the smooth grassy greens used in lawn bowling, a rub was a bump or uneven area that could send balls off course.)

Modern example:

"There's the rub. What does a progressive institution like Smith [College] do when Barbara decides to become Bert? It's a problem." — Roger Kimball, The New Criterion, May 2005