RealGM

I know that when the contract is up, the player becomes a rFA and the team has the right to match any offer to sign him without salary cap impact or something right? But does the player have sign with his original team if they decide to match the offer?

if they're a restricted free agent and their team matches the offer, then yes they do have to sign with them. When restricted they can only sign with another team if that team makes an offer their current team isn't willing to match.