Meaning of the final room in 2001: A Space Odyssey

The ending is largely symbolic, and reading the book helps with its interpretation. Here are some different opinions. The one based on the book goes like this:

The black monolith you saw in the beginning of the film, and the one found buried on the moon, were placed there by a supremely advanced alien race as a way to nudge along the evolution of humanity. The first encounter with a monolith gave our ancient ancestors the knowledge of how to use tools.

Next, the one on the moon pointed us farther out into space, toward Jupiter (and may have given us the ability to create artificial intelligence i.e., HAL, which was another massive leap in technology).

Finally, Dave reaches the third monolith and is transformed into the next stage of human evolution, the Star Child seen in the final shot. The psychedelic light show before that represents the vast knowledge being given to him by the monolith, which our minds can't even comprehend (and also, possibly, him being transported across enormous oceans of time and space).

Another source gives something similar:

After Dave points to the monolith he is transformed into the Starchild which is a glowing, fetus like being but also has the appearance of a man in the face. After this scene Dave is sent back to Earth by the monolith and the film ends as Dave, now the Starchild, looks on at our planet. The reason the Starchild was sent back to Earth is because the monolith has completed its duty of advancing human life to its evolutionary limits so it sent the Starchild back to continue its work. In the last 30 minutes of the film Dave is given all the knowledge of the universe and has been transformed into an immortal, all knowing being. Dave is now the film's version of God sent back to advance his race further than they ever could have by themselves.

In other words, the room is a place where Dave can observe our planet and play God in order to further advance the human race.

However, Kubrick himself admits that

you're free to speculate as you wish about the philosophical and allegorical meaning of the film

so there may not be a definite answer. Some theories even say that the screen itself is a monolith, where the audience is drawn to.

In the book, Clarke expertly shows that the Power behind the monolith does not want to hurt Dave in anyway. The monolith creates an environment for Dave to exist in that would not harm him in any way, makes it look like a hotel room, filling it with familiar items to assuage any fear and appear welcoming.

Dave can't believe what he sees, but leaves the pod and explores the room in his suit. He sees the telephone and telephone book, but the phone doesn't work and the telephone book is blank.

He explores more and finds the refridgerator, where there is a variety of packaged food, but it is all "blue substance, of about the weight and texture of bread pudding. Apart from its odd color, it looked quite appetizing."

There are clothes in the closet, which are a bit out of date for Dave's time.

Dave decides to trust the enviroment. "But this is ridiculous, Bowman told himself. I am almost certainly being watched, and I must look an idiot wearing this suit. If this is some kind of intelligence test, I've probably failed already. Without further hesitation, he walked back into the bedroom and began to undo the clamp of his helmet. When it was loose, he lifted the helmet a fraction of an inch, cracked the seal and took a cautious sniff. As far as he could tell, he was breathing perfectly normal air."

He eats the blue food and drinks the water which "..tasted terrible because it had no taste at all; the faucet was supplying pure, distilled water. His unknown hosts were obviously taking no chances with his health."

He showers, dresses and he turns on the television and "All the programs were about two years old. That was around the time TMA-1 had been discovered, and it was hard to believe that this was a pure coincidence. Something had been monitoring the radio waves; that ebon block had been busier than men had suspected."

Refreshed and exhausted, Dave lies down on the bed, turns off the light and "...for the last time, David Bowman slept."

The Power behind the monolith then transforms Dave into the Starchild, the next evolution of man.

In the movie, however, Kubrick shows Dave growing old in the room and dying a natural death. Whether the monolith manipulates time, or Kubrick decided to show the passage of time using the scene cuts where Dave ages, is up to opinion. But Dave ages and dies and then is transformed into the Starchild. Again, however, the room is created as an environment for Dave to exist in prior to his transformation.

A recently unearthed interview with Kubrick brings more light to the ending scenes of the film.

The French room is basically a zoo enclosure in which Dave was studied before he was transformed into the Starchild.

Source

"The idea was supposed to be that he is taken in by god-like entities, creatures of pure energy and intelligence with no shape or form," Kubrick explains in the tape.

"They put him in what I suppose you could describe as a human zoo to study him, and his whole life passes from that point on in that room. And he has no sense of time.

"They choose this room, which is a very inaccurate replica of French architecture (deliberately so, inaccurate) because one was suggesting that they had some idea of something that he might think was pretty, but wasn’t quite sure. Just as we’re not quite sure what do in zoos with animals to try to give them what we think is their natural environment.

"Anyway, when they get finished with him, as happens in so many myths of all cultures in the world, he is transformed into some kind of super being and sent back to Earth, transformed and made into some sort of superman.

"We have to only guess what happens when he goes back. It is the pattern of a great deal of mythology, and that is what we were trying to suggest."