Start Gently

With any deposit on a surface, the first step is to assess how durable the stained or deposited surface is. Since we are dealing with a wet erase gaming mat, I’d recommend starting with the gentlest solution and moving upward. To that end, a bit of warm water with a few drops of a biodegradable soap (Dawn is a good brand, but any biodegradable soap is fine) and a dry micro fiber cloth is where I would start.

Take the cloth and dip a corner of it into your soapy water, and try working out the dry erase marker in a small, isolated corner of the mat. If this works, continue around the mat - refreshing the water or cloth as needed. Sometimes a little bit of force is required, but you should not have to struggle with it; if it works, it will be apparent relatively quickly.

More Power, More Gently

It is worth noting that in the long history of stains and deposits, two solutions available have almost always relied on using an acid or a base. This is not an ironclad law of stain and deposit removal, there are certainly other reactions which precipitate removal, but it is a very handy tip to keep in mind.

Should your warm, soapy water fail to remove the stain, it might be time to try something acidic or basic.

I prefer starting with an acid. If you have white vinegar or lemon juice, either of those are gentle enough acids for this purpose.

Again, get some warm water and add about a quarter as much vinegar or lemon juice to that water (such that a generally 4:1 ratio of dilution is achieved), and, again, a microfiber cloth. Take a corner of the cloth and wet it with your acid, and, again, make your first attempt in an isolated area of the mat.

Do not let the acidic solution sit on the mat for more than a moment or two before wiping it up with a warm, soapy cloth.

If the acid fails to remove the deposit, it is time to try a base. Take a moment to thoroughly wipe up any acid that may be lingering on the mat.

Now, repeat the above procedure, but replace an acid with a gentle household base, such as ammonia. Be sure to dilute the ammonia and always be sure to first try the ammonia dilution in an isolated area of the mat, and be sure to fully wipe off any ammonia or other base residue.

Nothing Gets Chocolate Out... See?

Sometimes, however, a deposit will not be removed without overdue effort and/or damaging the original material. Should the above three methods fail, I leave it to you to decide how much you are willing to invest in the quest to remove the stain. This answer attempts to provide some clear instructions on methods to remove a deposit while also maintaining the original product’s integrity.

A Final Note

At the recommendation of user @KorvinStarmast, I am adding my followup comment in the main body of this answer. I recommend not using solvents unless you have exhausted all other options and are willing to lose the game mat itself.

If the decision is made to pursue the use of strong solvents (such as alcohol or even petrol-based cleansers) I strongly advocate using a 100% cotton, un-dyed terrycloth. Many microfiber cloths are composed of dyed polyester, which may be adversely impacted by the use of strong solvent materials, leading to the microfiber itself staining, scratching, or otherwise marring the surface of the mat. Despite cotton being less "grabby" than microfiber, it is also much more resilient.