I'm using CLR with a hose and scrub brush but it's not taking much off. When it dries the stains come back much the same. How should I clean this?
Muriatic acid (hydrochloric acid) is the best concrete cleaner. CLR is (mostly) phosphoric acid and soap. The problem is, concrete is essentially calcium carbonate which is just holding aggregate/material together.
Calcium carbonate will partially/slowly react with phosphoric acid to produce calcium phosphate... which is probably what the "white stains" on the floor are. Muriatic acid will quickly react with calcium carbonate or calcium phosphate to produce calcium chloride. After cleaning you will probably want/need to seal the concrete again.
Edit- foam puzzle mats are EVA foam (or Ethylene-Vinyl Acetate copolymer). EVA foam is made using acetic acid (plus oxygen and ethylene). And then the "finished" pellets are rinsed with more or other acids (maybe phosphoric acid, although an acetic acid wash would be more common). Then the pellets are heat melted into mats or other objects. There's actually a good chance that there is some phosphoric or other acid left in the EVA foam, that's causing the staining (reacting with concrete to produce calcium phosphate).
Yeah those mats have should been rinsed clean* by the manufacturer, but costs are against us all.
A favorite trick of mine with surface stains on concrete, is to take a concrete block and crack it into chunks with a 5 pound hammer, or something. Then take a chunk of the concrete block and place the flat side down (what was formerly an exterior face of the block) on the concrete and start sanding in a circular motion (aka Mr. Miyagi's wax on/wax off).
Try it on a small inconspicuous area of your concrete and see what you think.