I made a trade deal with another civilization to get a luxury resource (I traded a silver, which I have 2 of, for truffles, which I don't have any of). However that didn't seem to increase my amenities at all and on the resource report I don't see any difference from trade. Even when I tried to do a deal where I bought a luxury resource for gold, I didn't see any change. What is going on here?
TL;DR Changes in amenities won't show up until you hit the next turn button.
In Civ 6, each luxury gives you 4 amenities, and extra copies of a particular luxury don't give you more amenities. According to page 50 of the manual:
Only one source of a specific luxury provides amenities, shared with the four neediest cities in your empire.
You don't have precise control over which cities get which amenities, and "neediest" isn't defined anywhere, so I don't know exactly where those amenities are going to go.
In addition, it often takes to UI a long time to update things. In this case, the engine that distributes amenities among your cities only runs after you click end turn, so the UI will definitely not reflect the new one until the next turn.
The example DuckTapeAl gives is wrong.
When you have 8 cities and 2 silver, only 4 of your cities will get one amenities, the 'extra' silver is useless and won't do anything. However, when you trade like in his example, every city will get 1 amenity.