A while ago, the only pegs available for mulch mats were made of plastic. We really didn't want to sell compostable mulch mats with bits of plastic that would stay in the ground for centuries. So we used cocktail skewers which did a reasonable job but were a little flimsy and could be faffy to insert. We still have these in the wildflower seedmat kit as they are perfect for small projects like this.
However. I am delighted to say that now there are proper, robust bamboo pegs available. These can be stamped or hammered in as necessary. These are particularly useful with our larger square mats, rolls of garden felt, and rolls of jute matting.
Quite often, what we find is that once it gets wet, and grass underneath sort of grows into it in an attempt to get through, it makes it adhere to the ground and the material it's mulching contributes to the mulching properties of the mat. So they don't necessarily need pegging down. But it makes a neater job of it, especially in locations prone to wind.
Simply arrange the mat around the base of the tree. You may need to stamp down any weeds or grass that's already protruding. If it's long don't bother to cut it, just trample it down. Then peg each corner down with a bamboo skewer. There are 40 in a pack, enough to go with up to 10 of our 100cm x 100cm mulch mats.