If you like painting standing up rather than sitting down, a palette is a very useful tool. It enables you to have your colors (and brushes) at an ergonomic working height and it's instantly accessible when you want to pick up a color or mix a color, whether you're standing in front of your painting or viewing it from a distance.
If you're using a traditional wooden palette, a plastic one, or a disposable paper one as in the photo, the principle is the same: use your forearm to support the weight of the palette. This stops your wrist having to hold the weight, and is far less tiring.
Insert your thumb through the hole, then curl your fingers around the edge or just rest the palette on top of them. Hold it firmly, but not in a panic grip. You don't want to get a cramp in your fingers, you just want to be sure you don't drop the palette when you put brush to paint.
Be careful you accidentally don't lean into the paint on your palette. It's easy to do if you bend over to pick up a brush you've dropped, for instance.
If you use your wrist to hold up the palette, with the bulk of it floating out in the air, you'll get tired far more quickly. The palette also wobbles more as you pick paint off it with a brush, or mix colors on it.
That said, you could rest the other end on your easel or table. This takes the weight off your wrist and gives more stability.
If you like to paint with multiple brushes, you can hold these in your fingers together with the palette. This means they're immediately available to use, without bending or stretching to reach them. It also solves the problem of where to put a brush with paint on it without marking a surface.
You can hold one or two brushes or a whole bunch together with a cloth for wiping a brush. Your dexterity is the limit. You'll find that with practice you stop consciously "holding" the palette and it'll just rest in your hand.