Welcome to Fluther.
Yeah, that looks like hell, all right. Depending on the amount of corrosion there, and whether it has affected the sink itself, you may need to plan on replacing the sink, too. Just keep that in mind.
Start by disassembling the PVC (plastic) drain pipe at the strainer. I’d recommend having an empty bucket handy, because there will be drain water in the P-trap below the sink (since that’s its whole purpose), and you can avoid dumping that on the floor. (Depending on how long you’ll have the plumbing disassembled, you should also have some rags around to stuff in the open drain pipe. Plugging that serves two main purposes: 1) You won’t lose anything down the open pipe, and 2) Even more important, you’ll prevent sewer gases from escaping into the kitchen, which can be a real hazard. Be sure to stuff the rags so that you can pull them out before you reassemble the pipe!)
Hey, be sure not to dump that water down the sink drain after you collect it. Don’t ask why I would give such obvious advice, as if you’re as stupid as… well… as stupid as someone who has done that. So just don’t ask.
Once the PVC pipe is out of the way, and again depending on how bad the corrosion is of the sink itself (and whether any of that gunk is adhesive intended to stop the leaking), you may be able to find the threaded ring that secures the strainer beneath the sink. That has to be loosened for removal, so that the basket will simply lift through the top of the sink when the fastener is removed at the bottom (You can’t even see the fastener in the photo.)
It would be worth a trip to a hardware store to see what your replacement parts will look like and how they’ll fasten (if you don’t already have them).
I expect you may have to chisel, grind, chip or otherwise mechanically remove that gunk before you can get to the fastener ring on the basket.
When you replace the new basket, be sure to follow the instructions to use a bead of plumber’s putty to seal the thing. If that had been properly done for the installation we’re looking at, then we wouldn’t be looking at it. This joint should never leak.