A "tray" is a flat dish or container that's used to carry or serve food. ... Most trays hold food, but they can also be used for carrying, storing, or displaying all kinds of things. In the case you listed the "tray" is what the shot glasses are being carried on.
"made more rounds"
First you have to understand that in the sentence a person is carrying around a "tray" that has filled shot glasses on it. They are going back again and again to the people wanting drinks (shots).
To help you understand this more I want you to think about going into a bar and ordering a drink. Once you have finished with it, you decide to order another. Each time you order you get another "round".
Hope this helps.