How to remove old single-hole kitchen faucet
Some interaction between the center threaded rod with the slot in it and the outer tube is what holds "C" clamp in place. I would try holding the center rod with a large flat blade screwdriver in the slot and loosening the tube with pliers by turning the rod CCW. If that doesn't work, then try turning the rod CW and see if this releases the outer tube.

The outer plastic nuts just unscrew.