thank you

When a person is working or on duty, to continue on their rounds means to return to work. Example: After lunch, the security guard continued on his rounds, and inspected the building.

A waitress has to check on all of the tables that are her responsibility. The process of checking those tables is called, "doing her rounds" or "making her rounds". From the context given, it seems that she stopped at one of her tables and had a conversation with the people at that table. When the conversation was finished, she continued to check the rest of her tables... "continued on her rounds".

In a hospital, doctors or nurses who are on duty have many patients for whom they are responsible. Making one "circuit" of these patients where each is medically checked, is called "a round". So, having a series of customers who you tend to sequentially is called making the rounds. The waitress made the comment and went back to serving her customers.