My friends think you can add it to you hand, and play some other card from their hand.

A: You are correct, your friends are incorrect. You pull the top card off the deck, and immediately play that card. You do not get to add it to your hand, or play any other card from your hand.

Note that you may be seeing this FAQ because your deck has a version of this card with a different name. There are so many analogues for Mystery Play that you should simply check the Fluxx Comparison Chart. Highlight the row for the Rule: Mystery Play, and scroll across to the deck you have to find the card this question refers to.

Note that there are a couple of Keepers which grant the owner the personal ability to do a Mystery Play. Those are also listed on the Fluxx Comparison Chart, below the Rules, Actions, and Surprises.
Jumanji has The Game
Marvel has The Infinity Gauntlet
Star Trek: DS9 has The Wormhole