What are the levers on my washing machine hose connection fittings?
Those knobs are indeed quarter turn shut off valves. They have probably not been turned in years and have buildup of mineral deposits inside or some other type of corrosion.

The shutoff valves are there to allow it to be easy to attach / replace the washer water hoses. 

Before trying to see if you can free them up you will have to figure out how to turn off the water supply up stream from where these are located. You may be having to replace the whole water hammer arrester assemblies to get valves that will work. 

Note that it is quite common that those in wall laundry hose hookup boxe units having crappy shutoff valves. The ones in my house are similar but have a simple round turn knob on the top and are very difficult to turn after sitting for a long time.