It’s handled by pict-rs, an image hosting server than runs next to lemmy, which lemmy interacts with. When you upload an image on lemmy it goes to pict-rs, and you get a link to that image which is then used for the post or comment.
You can use any other image host that provides a direct image link, as well.
It also worth noting that most instances will host cached thumbnail copies of the images in all posts, regardless of what instance is the main host, or even if using an external host entirely.