Push it out from the other side, or put a screw in it from this side and pull it out. There are different kinds of inserts with different types of barbs so it might splinter the wood.

An alternative is to drill it out.

Drill them out with a slightly larger drill bit then fill the hole.

If you really just want to pull it out I suppose you could find a screw that matches this nut and then you will have something to pull on. I’d be worried what kind of damage to the surrounding wood that would cause.

I’m by no means an expert, but off the top of my head I don’t think I’ve ever seen a purpose made wood anchor, threaded insert, t-nut, etc. with a hex head like that. They may exist, but I’m not personally familiar with them.

Which makes me suspect that what you’re looking at is just a regular nut stuck into a hole that’s been carved into the wood, and probably glued into place, so my suggestions are going to be based on that assumption.

I would try to find a matching bolt that you can grab onto to pull it straight out, maybe use a prybar or claw hammer for some extra leverage.

It’s possible that a little heat could break the bond if it’s glued in (depending on the type of glue.) You of course need to be careful with that, but you could try carefully using a blowtorch, heat gun, or lighter. If you have an inkling about what kind of glue was used, you can also try to find a solvent that would work with that.

Failing that, if you have access from underneath depending on how the size of the hole in the insert matches up with the through-hole in the wood (assuming it’s a through-hole, of course you could always make it one if needed) you may be able to get some kind of punch at it from the underside to hammer it out.

If that doesn’t work,you can try drilling it out. Personally I’d start from the bottom and see if that would allow you to hammer or pull it out, then fill the hole with a matching diameter dowel glued in place and cut and sanded flush. You could also grill out the nut itself, but may take a while, and may wreck your drill bit if you have a cheap bit.

Depending on how flush it already is and what kind of finish you’re going for, you may just be able to slap some putty over it and call it a day.

You could take a threading tool that is reverse threaded and just twist it out.

If the wood will be painted it might work with a ‘hole saw’. Saw a rond hole removing the anchor and then saw the same size hole in a different piece of wood giving you a wooden disc. Glue the disc in the hole and sand it is completely until you can’t feel it anymore. Paint it so it is also not visible.

Do be aware that this is very hard to do and it might not look as nice. Try it on some scrap first.

Yo could try to look for some screw removal bit, https://www.eastwood.com/eastwood-6-piece-screw-extractor-set.html

I had this problem once and I ended up drilling it. Fortunately my drill bit is harder than the anchor, so it works. Wear eye protection though because metal dust is no joke.