What am I supposed to do when my chain falls off? It happened when I "stepped on it" (on an empty residential street). It left the chain on the road.

What tool(s) and parts should I have had with me, to fix it?

It was replaced (new) two months ago, since when they also had the back wheel off to replace its brake pads.

When I lubed it recently I noticed that one of the links seemed loose, but I didn't want to take the time to take it back to the LBS (which is now on reduced less convenient winter hours).
A chain tool (aka "chain breaker") for sure. Many multi-tools include one. Depending on the chain, also a quick link (they come by many names) or special pin.

Basic process is use the chain tool to remove a pin or two so that the mangled chain bits can be removed, then reassemble to a shorter chain. Try to leave a pin still in one outer plate; it's a lot easier. When you reassemble it, do it on the bike threaded through all the derailer bits properly.

I've done this roadside before. Took me 5-10 minutes, some cursing, a little fiddling with the tool, and ended with very greasy hands. A few of those minutes were waiting for a crosswalk signal so I could retrieve the chain.

Make sure to avoid the big/big gear combo or even your big chainring entirely, until you've replaced the chain. And replace the chain right away.