How do I replace my faucet with these connectors?
Well, while we wait for the question to be completed... those things pictured are plumbing tees on your water lines, presumably in your basement.  The technology is this "new" plastic called PEX.  

But that is all besides the point - Normally your supply lines run up into your cabinet and to a local shutoff valve.  The sink faucet normally connects to the shutoff valve.  If your laundry sink is in the open, then you should trace the line to the faucet.  Looking for some valve.  If you don't find one, curse at the plumber, and plan on cutting one into this line close to the sink, after you shutoff your main.  

Plumbers that do PEX often have specialized tools which are more that homeowners want to invest in.  Ask at the store for simpler connectors.