There is no one way slaves interpreted the bible. You have to realize that the slave community was not monolithic in terms of theology, convictions, experiences, and the like. Therefore, there will be differences in emphasis. That said, we can make a general point. The person who has done the best work in this area is Orlando Patterson in his book Slavery and Social Death.
In one section, he argues that the slave owners and slaves were reading the same bible, but they were emphasizing different things. For example, all loved the apostle Paul. However, they emphasized different aspects of his teaching. The slave owners emphasized his writings on order. Slaves emphasized Paul's teaching on salvation and deliverance.
In a nutshell, most slaves interpreted the bible as a book of salvation from oppression. Some of the dearest passages were passages like the Exodus narrative and anything that had to do with heaven.
From a theological perspective, slaves emphasized what theologians called eschatology, that is, theological rubrics that deal with last things. Here is old hymn that sums up this point:
Deep River, My home is over Jordan. Deep River, Lord. I want to cross over into campground. Deep River. My home is over Jordan. Deep River, Lord, I want to cross over into campground. Oh, don't you want to go, To the Gospel feast; That Promised Land, Where all is peace? Oh, deep River, Lord, I want to cross over into campground.
I would like to weigh in on a few points regarding slaves and the Bible. First, it should be pointed out that in most places, slaves were not permitted to learn to read or write, so most could not read the Bible at all, but only have it read to them. They may have appreciated its beauty as they heard it, but they could hardly take it home and study or discuss it, as could people who could read. Only that which could be memorized could be mulled over later. Second, we have no way of knowing, really, what parts of the Bible were read to them, whether it was read to them in its entirely or whether sections were selected such that they were meant to support the idea that white people were their masters. Certainly, sections of the Bible are often chosen to make whatever point the person selecting wishes to make, with a complete disregard for other sections that might contradict that point. Finally, what I do not think has been considered a great deal is that slaves, at least the first ones brought over, were from Africa, and they brought whatever religions they had had in their respective homelands. It is difficult to know what they made of this new religion and how or if they reconciled it with their former religions, forsaking old ideas for new ones, trying to make the new ideas consistent somehow with the old, or maintaining their original religions while pretending to go along with what their masters wished.
Many of the spirituals that the slaves sang were derived from the Old Testament, such as Go Down Moses and Deep River. Also, as mentioned already, the Book of Exodus in the Bible was interpreted by the slaves as reflective of their own conditions, and many named their children after personages from the Old Testament. So they put themselves in the place of the Jews who were enslaved under the Pharaoh using the story of the Israelites as a blueprint for their deliverance from their owners. By connecting themselves with the Israelites, the slaves also created a sort of mythical past for themselves and an identity to which they all could relate and under which they could all unite. Moreover, this connection provided the slaves with a hope that they, like the Israelites, would someday be released from their enslavement.
There are many different ways that the Bible was interpreted by slaves in the South. However possibly the most significant interpretation was that regarding equality. Obviously slaves were not considered equal in the South during their enslavement. However the Bible stresses that all are equal in the eyes of the Lord; this concept brought up many questions and challenges to slavery. If everyone is equal in the eyes of the Lord, then why were blacks being enslaved and being considered lower than white people? This is an important question that the slaves asked themselves, leading many to confirm their belief that they should not be enslaved or treated differently than anyone else.