Here's an analogy that I think will be easier to understand because it uses familiar objects and units.
Imagine you have a pressure washer that needs 10 gallons of water per minute. You have a well with a pump that produces 2 gallons a minute. To run that pressure washer off of that well, you're going to need a storage tank and to store up some water. That storage tank needs two things: first, it has to be big enough to hold enough water for the amount of pressure washing you intend to do. Second, the pipe coming out of that storage tank has to be able to provide 10 gallons of water per minute. Let's say you get a 100 gallon tank, you could pressure wash for ten minutes and then you'd have to wait 50 minutes while the tank refills. If you put a one-inch pipe on the outlet of that tank it could provide 10 gallons per minute until it's empty.
In this analogy:
The well is the alternator
The pressure washer is the winch
The storage tank is the battery and the outlet pipe is the cca rating of the battery
Amps are a measure of flow, it's analogous to gallons per minute. Amp-hours are a measure of capacity, analogous to gallons. Multiplying flow (amps or gallons/minute) by time gives capacity (gallons or amp-hours).
So let's go over the numbers:
The winch has a maximum draw of 390 amps. This is analogous to the pressure washer needing 10 gallons per minute.
The battery is rated 850 cca (cold cranking amps). This is the battery's ability to provide flow, analogous to the 1-inch pipe. This battery has the ability to power this winch at maximum draw.
The battery has a capacity of 1100 amp-hours. This is analogous to the capacity of the tank. Theoretically, it should be able to power the winch for over two hours (but see below).
The alternator is capable of producing 55 amps. This is analogous to the capacity of the well. This means you can operate the winch 8 1/2 minutes out of every hour without draining the battery. Alternately, if you operate continuously and drain the battery it will take 20 hours to recharge it.
Finally, there is an important difference between batteries and water tanks. If you put 100 gallons into a water tank you can take 100 gallons out, and it doesn't matter how fast or slow you take that water out. Batteries don't return all of the energy put into them, and the faster you take it out the less you get. The 20-hour rating for amp-hours is the amount of energy you get by evenly draining a battery over 20 hours, which is much slower than most real-world applications. A rule of thumb for estimating capacity for higher-flow applications is to take the 20-hour rating and cut it in half. So in this application I would say one hour of continuous use (followed by 20 hours of charging) is your outer limit.