It's up to your interpretation. The "black projector screen thing" is actually a large black obelisk, the same kind that was buried on the moon and that appeared on Earth "at the dawn of man" at the beginning of the film. The easiest interpretation of the black obelisk is that it instigates or accelerates the evolutionary process: for the ape-men, it apparently caused them to develop the concept of tools (which they then used for warfare.)
For the astronaut Dave, his presence at Jupiter (the supposed origin of the obelisk) apparently caused him to accelerate through space and time, living out a personal progression in his own little mental world represented by the mansion, then finally dying and being reborn as a more advanced, more highly-evolved form of human. I think the common interpretation of the fetus at the end is that the obelisk, using Dave as its template, has once again advanced humanity forward and that the new form of human being is soon to arrive on Earth.
The movie is based on a novel by Arthur C. Clark. I haven't read it, but doing so might give you more insight into the meaning of the film's abstract ending.