All of this is done in memory of the exodus of the Jews from Egypt. The story is that Moses and God brought plagues to convince the Pharaoh to free the Jews, who were slaves at the time. The last plague was that all the firstborns in Egypt would be killed by the Angel of Death. However, the Jews were told to cover their door frame in lamb's blood, and the Angel of Death would see the blood and pass over that household and spare the firstborn. After that, the Pharoah let them go, but they had very little time to exit Egypt... so little time that they weren't able to let bread rise or leaven (about 18 minutes).
My experience with Passover so far makes me think that a big focus of the feast is on the fact that the Jews had to exit Egypt quickly. I shared this observation with a friend of mine who is a Jew, and he asked what else I thought it should be about. "The passing over of the blood stained door frame! Hence the name Passover." He then said that that part of the story isn't hardly focused on in the Jewish Passover feast anymore. Interesting!