Idk if this is the textbook answer, but what I personally use them for most often is for catching fluids during a washout.
Say your thigh has a nasty infection - like a deep wound, pus all over, tissue is dying, not a good time. So you get sent to the OR, where we'll irrigate the fuck out of it, trim out all the tissue that looks like it's dying, then irrigate the fuck out of it again, then irrigate some more. We're talking like 5 liters of saline being shot into your wound and allowed to just flow right back out. If we let that just hit the floor, we'd be ankle-deep in a coctail of saline, blood, pus, and whatever pathogen is causing the infection, so enter the kidney basin: the curve shape allows it to conform to the curve of your leg, so if I hold that below your wound while the surgeon is blasting it with saline, it'll catch the mess before it hits the floor. I'll generally hold with with my fingers wrapped around the bottom of it and thumb around the inside of the basin; that way I shove a suction tube between my thumb and index finger so that the tip of it is resting at the lowest point in the basin. So, as nastiness falls into the basin, it's immediately sucked up out of our sterile field.
Other than that, I like to put all my fine/sharp instruments in the kidney basin at the end of a case, shove the basin full of sharps into a side of my instrument set, pack the other instruments around it, and send them to our sterilization department like that. That way whoever gets my dirty set in decon can very clearly see where all the sharps are, so they don't get stabbed when they're sorting out and wash all the instruments. Some techs will just dump all the dirty shit into the tray with no logic or organization, and it all just looks like a bloody metal birdnest, which is a fucking nightmare to deal with in decon (I used to work in sterile processing, so I know the pain).