Yes, it is a very formal, and, posh phrase :P
I personally can only think of the following situations:
(1)a waiter/waitress saying 请慢用 to the customers when the food is served (I think this is the most common situation in whicn you would hear this.)
(2)to important guests when you treat them at a fancy/posh restaurant, when the food is served and you would like to suggest starting the dinner, such as to your business clients etc.
(3)I agree with Wang Haoyu. If you have to leave while others are still having dinner, especially when you are not familiar with them/meet them for the first time, you could use "不好意思，我得先走了，你们请慢用" to tell them that they don't need to hurry, just enjoy the meal and you feel sorry to leave earlier.
I would not say this to those who I'm very familiar with.
For example, I would never say this to my parents no matter where we have dinner.
I would never say this to my friends either. Or I could, if I wanna be humorous, I guess.
This phrase suggests great respect to the guests. However, such respect can imply that you are not familiar with each other yet.That's why there is no need to say this between friends and family members.