Mona: I am so happy that you could visit me. Raul: Me, too. ( = I am happy that I could visit you.) Mona: I want you to make the most out of your stay. Raul: What do you mean? Mona: I will not let you just sit around in my house and watch TV. We will go places during the five days that you are here. Today, I am taking you to a museum; Tuesday will be a trip to the nearby mountains; Wednesday, we will spend it at the beach; Thursday, we will take a bus tour to show you the beautiful houses where the rich people live; and on Friday, before I drive you to the airport, we will spend the morning at an amusement park. Oh, yes, we will also take a short train ride to visit some famous vineyards. I will get you to the airport in time to catch your 9:30 p.m. flight home. Raul: Wow! What a busy schedule. I really will make the most out of my stay, won't I!

You will be visiting another place. To gain the most enjoyment from your time there, there are certain things which it is felt that you should see and do.

It means to make sure you get as much enjoyment out of it as possible.

To "make the most of your stay" means to enjoy your stay as much as you possibly can. Did you hear this in a hotel message? Hotels may wish to help you "make the most of your stay" so you'll enjoy yourself fully while staying in their facilities.