What is this thing that holds widespread bathroom faucet handle to sink? How do I tighten it?
This is a faucet that is designed to install from the top only.  Price Pfister calls it Top Pfit.  The piece you have pictured was probably over tightened, bent, and popped off.

Update: It looks like the old name for this was "Twist Pfit" back when they used it for bathroom pfaucets.  Top Pfit seems to be a new name they use for kitchen pfaucets.

It works like a toggle bolt.  It lines up with the pipe to slip into the sink hole then it flops down to a horizontal position where you use a nut or screw on the top side to tighten it.

If you remove the handle from that side valve, you should be able to figure out what to turn to make the carrier with the studs on the side move up and down.  Once you figure it out, the toggle will need to be bent back into shape and snapped back in place.  DO NOT over tighten the valve once you figure it all out.  You can use plumbers putty or even silicone if you are desperate to hold it in place semi-permanently, but over tightening will only get you back to where you are now.

If you look through the PF site, you might find some installation instructions close to your faucet if you need any help, but the design may have changed slightly over the years.

Pictures from found install manual: