That appears to be a standard PVC ball valve. Take a trip to a plumbing supply house or a big-box store, and grab a couple different PVC fittings and pieces of pipe that measure as reasonably close as possible to the internal size of the open valve end. Use a tape measure or internal/external caliper to match the diameters.
When you get back home, find the fitting or pipe that press-fits reasonably firmly into that ball valve end. That's the one whose pipe size you should use to come out of the ball valve.
Attach a small piece of the correct size PVC pipe to the ball valve, using standard PVC primer and glue. Then attach a PVC union:
of the same size pipe to allow removing the drain hose from the ball valve. The other end of the PVC union can be attached to a length of rubber hose, through a hose barb fitting. Hose sizes of the same size as the ball valve's PVC pipe are available, but may be a bit expensive. You may want smooth-inside pipe instead of corrugated pipe, so the stuff that needs filtering won't catch and clog in the pipe. Fittings exist to go from pipe to the hose, and hose clamps can make the hose grab the hose fitting well. The PVC union has its own spinning lock nut, so the whole hose won't have to turn to remove the hose from the ball valve.
At the end of the hose near the yard drain, you can get a mesh laundry bag with mesh size less than the chunks & stuff that will come out of the backwash. A large enough bag can catch all the junk, or you might have to swap out bags. Attach another hose barb fitting to the yard drain end of the hose, so the hose will have a lip, and the bag won't slide off the end of the hose when the water runs.