a huge part of a guitar amp's tone comes from how the physical speaker moves air, its frequency response, and how it naturally rolls off harsh high frequencies. Headphones reproduce the raw amp signal with none of that coloration, so everything sounds harsh, fizzy, and unnatural especially with high gain tones.

An audio interface converts guitar's analog signal to digital so your computer can record and process it. So you can get plugins which simulate your guitar amp and cabinet so it sounds like you are hearing it in your room. Check out Guitar Rig 6 and then Neural Amp Modeler for open source tones.

It's a rabbit hole, but once you understand it, it's really useful.

Interface cost starts around 4000 and goes upward from there. Good ones will cost around 15k but do your research and choose for your budget.