The name 'Skinner box' is a flippant alternative to B.F. Skinner's phrase 'operant chamber.'
Before the box, the prevailing approach to the study of animal behavior was the rat maze, which required the experimenter to move the animal many times, which usually got it excited and messed with the data. Also, the rat couldn't itself set how often it ran the maze. The box is called a free operant apparatus because, unlike those earlier methods, the animal can work the operanda (e.g., the lever) whenever it likes. This technique allows different topics to be treated, and Skinner explored a lot of them (for example, pioneering the study of schedules of reinforcement.) Another advantage is that the box is relatively standardized, which means it is easier to compare data across different boxes. Also, the box has been tuned, eliminating many annoying bugs. For example, in early versions the levers were not rounded, which consistently caused the rats to chew on them (hence closing the switch over and over again.) Skinner wasn't interested in studying this chewing, so he rounded the lever. Yet another advantage of the operant box is that it doesn't require you to code behavior, which takes forever and sometimes yields dubious data - with the operant box you just record responses. (This is also a weakness - like the earlier maze methods, you don't see that much about what is going on.)
But the most important advantage the box has is that it's really convenient. When you are actually running an experiment, you can set up your experiment to run while you're away, put the animals in the boxes and go have lunch instead of standing around and making them nervous. Before we had computers to record the responses, Skinner's invention of the cumulative recorder - a rotating drum with paper on it, and a pen which ticked up the paper every time the animal responded - made data collection an equally unattended process. If you need to collect days of data for hundreds of animals, this is a complete godsend. Too bad that not all kinds of research can be this convenient. Whatever you think of Skinner, he invented some useful lab gadgets, and many of them are part of the Skinner box.
Is the Skinner box hellish for the rat? Rats get nervous in the open, not in enclosed spaces - let one go and he will dart for shelter. The typical box is plenty big enough for the rat to walk around - it's not as if he's being kept in restraint. Besides, lots of people have pet rats which get morbidly obese in their cages anyway, doing nothing but eating and sitting around. I think playing an odd sort of game with levers is a lot better than that.
For a student not focused on animal behavior, the point of putting an animal in a Skinner box and watching it work is so you can learn something about the animal's behavior by watching and interacting with it. It's much more accurate and detailed than reading some oversimplified textbook treatment. Here you have the thing itself instead of someone's opinion. Here, even if you are the most skeptical person ever about behaviorism, you think everything they say is full of crap, you can put that to the test. You can see for yourself, and if you make a clever experiment then you can convince other people too. More likely you just get a taste of how much is there to be seen, how much you can do with the method, but also how little you know about why the rat does what it does.
The procedures "classical conditioning" and "operant conditioning" do completely different things, so it is not really an answerable question whether one works better than the other. It's like comparing a hammer and a screwdriver. Classical conditioning is glossed as the animal predicting and is effectively defined physiologically, while operant conditioning is glossed as the animal acting on its environment to produce an effect, and is defined much more broadly in terms of equivalence classes. (For example, we are concerned with when the rat presses the lever - but not whether he uses the right or left paw.)
If you think the process is merely trial and error, you should try making a robot which doesn't have any notion of how to press a lever built into it, but which can learn to do so from experience. Operant conditioning is easy for the experimenter because the animal is doing all of the heavy inductive lifting.