Frank Poole is killed while outside of the ship by HAL. David Bowman doesn't realize Poole is dead so he leaves the ship to try and rescue him. While he is getting Poole, HAL kills all of the hibernating crew of the Discovery One. After Bowman gets Poole's body he asks HAL to open the pod bay door so he can get in. In a famous quote ("I'm afraid I can't do that Dave.") HAL refuses and Bowman ends up going through the emergency airlock and risking his life.
He then puts on a suit to protect himself if HAL attempts to shut off the oxygen and goes to HAL's brain room to shut him off. HAL admits to Bowman "I'm scared Dave, I can feel it slipping away." and eventually loses all of its memory except for knowing a song called "Daisy" that his instructor taught him. He asks Bowman if he wants to hear it and Bowman tells him to sing it while he disconnects HAL.
After he does, he finds out that the Discovery One reached the orbit of Jupiter and he receives a pre-recorded message of from Dr. Heywood Floyd about what the mission was about. The crew was supposed to explore a new life form whose evidence was found in a black monolith that was buried on the moon, transmitting radio signals to Jupiter. The film enters it's final stage "Jupiter and Beyond" as Bowman finally sees the 3rd monolith and departs in a pod to see it. The monolith transports apparently through a black hole or worm hole through multiple dimensions and through different universes and galaxies and Bowman goes into a state of shock.
Suddenly, he is in a hotel room with no exit. He sees himself outside of the pod and suddenly the pod is gone and he is alone in the hotel room. He walks into the bathroom and sees that he has aged and his hair is gray. He then sees an even older version of himself eating supper acting as if nothing is wrong. The old Bowman at the table looks in the bathroom and sees no one there. He goes back to his meal acting as if he's lived in the room his entire life. He forgot the mission until accidentally breaking a crystal glass, which makes him wake up out of the trance for a second. He looks over at the bed and sees himself, only older to the point of mummification. The Bowman at the table is gone and the almost corpse one in bed is moments from dying.
He looks over and sees the fourth monolith at the foot of his bed. He silently raises his arm to it then falls back dead. He is then "reborn" into a supernatural being. A "Star Child" with the ability to travel through space and different worlds freely. The entire hotel sequence is a terrarium set up by the aliens to make Bowman comfortable and allow his "rebirth. The "Star Child" heads back to Earth and the movie ends with the Star Child outside of Earth looking at it with wonder that it is the primitive place it came from while "Also Sprach Zarathustra" blares triumphantly.