Hard to tell from the picture, but it's probably a thin underlayment, usually luan plywood or could be masonite, given vintage of fixtures. By tile, you mean vinyl tile? I don't know how successful sanding would be, but you can give it a try. The stained areas aren't currently wet?

I would be inclined to remove the underlayment, even if it's dry, to see if the subfloor is wet or damaged. If it's a longtime leak, then the floor joists might even be damaged. If the underlayment is wet, then definitely remove it, and figure out how to keep water in the tub. Plus that's not the right substrate for ceramic/porcelain tile.

I'd recommend using Ditra for your tile underlayment, and sealing it to the tub for waterproofing against future tub/shower spills.