How does one replace this part...?

While cornering sometimes, I hear what sounds like the tire rubbing on the wheel-well. Perhaps that's what's responsible for the breakage.

 

Besides the torn wheel well liner, it looks like you're missing the front wheel spat as well. Here's some pics, side, then front view, same area on ours:

 

If the tire is rubbing against anything, then something is bent or out of position, or somebody installed improper wheel rims or tires.

 

"Improper" tires is probably closest. The Michelin-X-ice are slightly oversized, but I need them for winter safety, so I'm not giving them up just because they rub the wheel-well once in a long while.

 

I had to include wheel rims because that was the source of rubbing on a previous car of mine. The steel winter rims offered by an area tire shop had the wrong 'offset' for my Subaru, pushing the tires out and rubbing during corning when the car was fully loaded. That was when I learned that Subies had different wheel offsets than domestic models.

Fortunately, the shop took the wheels back. Then I located proper wheels through what we used before craigslist arrived.

 

I'd probably have tried patching it using rivets before I spent the money for new parts, but that's just me.

 

Just marked up OP's pic a little, it looks like a portion of the wheel well cover is just gone, torn off (outlined). And there's a missing plastic fastener at the bottom (circled).

Doesn't seem like the wheel spat's there at all. Beside the spat there's another piece of plastic filling in the bottom that seems to be missing, it shows in the front view of spat pic I posted.

Look to see if the wheel spat bolts are still there?

 

If you can find the right patching material, drill a hole for a pop-rivet, fasten it, line up the rest and rivet in place. Not really too difficult.

 

Marked up my pic, the spat bolt locations, and that other piece, the plastic underpanel.

 

Bolts are stronger than pop-rivets, but harder to install.

 

Is this it?
TinyURL.com - shorten that long URL into a tiny URL

 

Well, all those bolt locations will have tack-welded nuts of some sort. With any luck the bolts are still there, just the plastic has torn out. Whatever, he can compare components from the other side.

Here's a better pic of the spat:

 

That is some exotic looking tread pattern. Snow tires?

 

They sure look like snows to me. Very similar to, but not identical to, the Michelin X-Ice3s on my Subaru.

 

Cool. Never seen one up close.

 

Yup, X-Ice2, a bit worn, down to around 8/32" tread depth remaining.

It's currently +8C in Coquitlam, think we've had one instance of snow this winter. We keep it away by all getting snow tires up here, lol.