I turned on the water in my kitchen sink, and the faucet popped right out, along with the water. I replaced it and tried to tighten the large bolt that it goes into, but it didn’t need tightening. When the faucet popped out, along with it was a small piece of black, rubbery, string-like substance. What is that stuff, and was it the only thing that kept the faucet from popping out?
You are being too vague. A rubbery substance is not used on any part under pressure to hold the faucet together. Only the seals would be a flexible substance.
Plumbers putty?
The remains of a washer, or “O” ring perhaps. It depends on the faucet design. It may be time to visit the hardware store and replace it.
It seems to be a rubbery, stringy substance *now. *I don’t know what it was originally. But it seems to have been the only thing keeping the faucet in place.
Sounds like it was probably packing string. Some faucets use that instead of O-rings.
No, it wasn’t the only thing holding the fauct together, or at least is shouldn’t have been.
Try reassembling the faucet without the string. If the packing nut won’t tighten then it’s either stripped or worn and you’re going to have to replace the entire faucet.
If it will tighten, remove it and wrap the string back around the faucet and tighten the nut back down. Replace the string if it’s frayed or worn.
I recently had a similar problem. It only needed a new O ring to hold it in place. 
The threads are what hold all the guts together. The sealing material doesn’t. This should be obvious, since it blew apart from the water pressure. You will likely have to buy a new faucet. Had it just leaked water the seal would have been bad, and not the threads. You could have replaced the seal, if the threads weren’t bad.
I can’t find any diagrams online at the moment, but there is a type kitchen faucet where the main spout is held in place by a compression fitting. The spout has a belled or bulged-out section at it’s base. An o-ring fits over this section and the whole thing is held in place by a compression nut. Without the o-ring in place, the bell section will fit through the compression nut. When the o-ring eventually deteriorates from use and age, the spout will pop right out of the base.
Short answer - you need a new o-ring.