Why does the prince still seem to be alive after being bitten by a snake and having gone to another planet?

This part can be a little hard to understand if you read the chapter just once, or too quickly. Try reading it a couple of times, slowly. 

Here's what happens in that chapter:

The pilot comes up to the little prince and is surprised and shocked to see that the little prince has been talking to a snake. Snakes in the desert can kill you in one quick bite, and they move fast.

(Now we...

This part can be a little hard to understand if you read the chapter just once, or too quickly. Try reading it a couple of times, slowly. 

Here's what happens in that chapter:

The pilot comes up to the little prince and is surprised and shocked to see that the little prince has been talking to a snake. Snakes in the desert can kill you in one quick bite, and they move fast.

(Now we have to remember here that near the beginning of the story, the snake had offered to kill the little prince anytime he wanted.) Although the narrator doesn't tell us exactly, we can figure out right here that the little prince was arranging to meet the snake later to be killed. The little prince is ready to leave the earth and wants to go quickly, without too much pain. Again, the narrator doesn't tell us right here what the little prince and the snake talked about.

The little prince and the pilot walk along together and talk for a while. When they come to the spot where the little prince has agreed to meet the snake, he walks forward alone, is bitten, and immediately falls over in death.

Whether or not the little prince actually goes back to his own planet after death is a matter of interpretation. The way I read the story (and the way that many readers do) is that the little prince simply died, and that the pilot takes comfort in merely imagining that the little prince has gone home.