What is this plastic twist in the back of my washing machine?
Without opening it up, it's hard to say for sure, but from here my crystal ball says...
It's the mount point for some internal plastic piece.
Since you didn't indicate that it felt like something broke as you were playing with that piece, it's not likely that you've broken anything. Since you say it doesn't fit as tightly as it did before, it might make more noise now as something is allowed more movement than it is supposed to have.
I'd suggest that you use your favorite internet search engine to search for &quot;Inglis &lt;model number&gt; parts diagram&quot; to find the parts diagram for your particular washer. Once you've found it, you'll be able to look through the drawings to see what piece connects to the back of the cabinet, and you'll know how critical it is.
You seem to indicate that it took some force to pop the metal cabinet out of the plastic piece, it will probably take a similar amount of force to get it back in. Maybe, by looking at the drawings, you'll be able to figure out how to jimmy it so it will fit snugly back into place. Of course, if it's not making any additional/unusual noises when it runs, it might not be that critical that you get it firmly attached, but since the manufacturer seemed to think that it was important to attach it here, it would be recommended to see if you can get it snugged back into place.
Take this as a learning lesson - don't go fiddling with bits of machinery just because! The parts diagrams to just about everything are available somewhere online - if you're curious, search for that first. Of course, if you like fixing stuff, fiddle away!
Additional Pro-Tip™: That little white plastic piece to the left of and above your arrow holds some other plastic internal bit in place. Unless you're feeling like tearing down your washer to fit it back in place, don't try pushing that clip back through the metal - it's there for a reason.