I use an old plastic credit card. Held at an angle across the mat, it works pretty well.

A little steel wool works well also.

I use a green Scotchbrite to remove mine. Scrub in both directions for the best cleaning.

I ditto the comment that a dull blade may have helped cause this. It has been suggested that a mushroom brush is good for cleaning this out. Also - it appears that your mat is older and dried out a bit. Olfa suggests that you periodically soak your cutting mats in warm water and let them dry flat - this helps keep the mat moist. I do mine in the bathtub where they can lay almost totally flat. Do not put in direct sun to dry since that might cause warping.

I use a damp washcloth, that does double duty of moisturizing the mat and pulls the batting right out.

Ditto the comment about a gum eraser. Works pretty well.

After reading this and HoneyJ's post, I took my really old eraser and worked on my mat. The batt marks are Gone! After scrubbing, I wiped over the spot with a damp cloth to remove eraser crumbs. I'm happy!

It looks like you don't use the grid lines for cutting so why don't you use the backside for cutting your material. I do that to keep my grid lines nice for when I really need to use them. It helps.

I use an old matt saved for just that purpose, and cut on the diagonal to avoid the ruts.

A friend made little scratchy pads using tulle that she cut into strips then crocheted. I use it a lot to remove batting or fusible fleece that also likes to leave bits behind now and then.