In the game, you draft flowers from a field in the row indicated by a farmer: up to three flowers if they're different colors and bee free; up to two flowers if they're bee free; or any single flower. Instead of drafting flowers, you can draft a pentomino from the 1-5 available, then fill it with some combination of flowers and hives, with each player having three hives. If you create a grouping of two or more same-colored flowers, you score points equal to the size of the group and record the creation of that color of honey; when you place another pentomino adjacent to those you played earlier, you can expand color groups (scoring previously placed flowers multiple times) or score new colors (with a decreasing bonus available to those who score all five colors) or both.
Once someone places their fifth pentomino, each player takes one final turn, then scores their hives, with each hive being worth points equal to the number of bees around it. Place bees smartly, and you'll score them multiple times. Whoever scores the most points wins.
You need to know a few other details to play Queenz, with those details covered in the video below, but the heart of the game comes from the tension between those who try to fill fields quickly and those who want to maximize points from colors and bees. Tied into that is your ability to control (to some degree) what other players can draft; each time you take flowers, you move the farmer ahead that number of spaces around the flower grid, and the next player can draft only from that row — which means you can (possibly) block someone from getting a color or being able to draft multiple tiles or being able to take bees.
I've played Queenz five times with two and three players on a mock-up copy from publisher Mandoo Games, with players staking out colors early, then trying to cut off one another, whether when drafting flowers or when taking pentomino tiles, especially when someone is about to puzzle together the perfect shape and flower/bee combos.
As you might imagine, the game is more cutthroat at two players since you each have more control over what your opponent can access. With three players, the farmer moves more between each of your turns, and the pentomino tiles vanish, then are refilled more often, so you need to be more adaptable and lean more toward helping yourself than trying to hurt others. After all, as all bees know you can sting someone only once before you die...