Sanding will always work, but just be careful as many chessboard squares are constructed with a veneer, so you're looking at maybe 1/128" or 1/64" max thickness. Don't start too aggressive like 60 or 80 grit, but instead start at a higher grit like 120 to be safer. If the top coat is flaky you can make quick work of it using a scraper, the kind with a knob where you pull on it to scrape like this.

If it was a larger piece like a table, or it had lots of grooves and valleys, a chemical stripper might be better but it's basically flat and relatively small so sanding or scraping would be best. Get it down to raw wood, then apply your favorite finish. Your final sanding grit will depend on the finish you use - just follow the instructions on the can.

There are a few chips on the edges you might want to address with wood putty or wood filler. It's not going to match perfectly but you can mask it a bit with those colored wood pens. If you have a table saw or are good with a hand plane, you can just rip off 1/8" each edge and then re-apply some plywood edge banding as the existing edge banding looks like it's in bad shape. They sell stuff you can apply with a clothes iron - super easy.

There are a number of furniture restorers on YouTube that do this type of restoration all the time. This video shows some techniques I described, from scraping, to sanding, to patching the voids.