To have "no reservations" about something means to have no doubt that something is good or right. To put in a reservation at a restaurant means to reserve seats at a restaurant. To say someone has no reservations can mean that the person did not save any seats. But, it can also mean that the person is saying that they have no doubts that this restaurant is a good restaurant. This joke is playing on the different meanings of the word reservation.
He recommended the restaurant to me , and said he had no reservations .