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At the base of the pin those round things are called barrel nuts. Essentially they just grip onto the pin on the back of the emblem.As 440 said, they can be a bit of a bugger to pry loose. Replacement are available on line, even on Amazon.
VERY CAREFULLY...unless you want to replace them with $$$ OEM or $$$$ NOS or semi $ repops. Use a #1 flat tip (small one) and carefully get under the edge, a little at a time, patiently working all the way around. When you get the flat tip in there, do not pry up, but just twist the flat tip a little to lift the emblem, just a bit. And work your way around. Do it wrong and the F and the 100 will break.
Or you can apply some propane torch heat and melt or soften the plastic barrel insert. Yes they are replaceable. That should release them a bit or make the emblem easier to remove. Yes once you get the emblem off, just push the barrel insert inside the hole and replace with new, after new paint.
Come install time, barrel insert 1st, then after you over haul the emblem (clean and repaint) get it started evenly all the way around and push in all at once with a flat item over the complete emblem. Like with a small piece of 2x4 wrapped in news paper. That is slightly larger than the complete emblem.