How do I get the metal rings around my hoodie strings off? (picture)?

Those are like grommets, they keep the fabric from wearing/ripping from the strings, they are usually crimped in place and probably hard to remove without tearing the fabric. If you don't want to see them, maybe you can just carefully paint them with dark blue nail polish or model paint or something like that?

I suppose if you absolutely have to have them gone, you might be able to carefully use a Dremel tool with a cut-off wheel to grind slots in it and pry it off bit by bit (watch out! the metal will get hot from grinding and might melt synthetic fabrics). You also might be able to carefully get a hold of them with small wire cutters and work it loose. Either way is going to be tedious and might damage the hoodie so you might want to practice of some old garment you don't want anymore and see how well it works.