Since the afterlife is a cornerstone of Christianity, this and the previous verse referring directly to the afterlife should be the focus of a lot of attention. However, like many of Jesus's statements, these verses don't lend themselves to a simple explanation, so they are often ignored or at least their complexities are.
Some ancient Jews (specifically Job) don't seem to have believed in an afterlife. It wasn't until Isaiah (Isa 26:19) that the resurrection of the dead is spoken of directly, instead of indirectly. The Hebrew word Olam Ha-Ba ("the world to come") is used to describe both the messianic age and the afterlife. Many Christians seem to look at "the kingdom of heaven" in the same way: the future world and the afterlife. However, Christ talks extensively about the kingdom of heaven and most of his statements are not consistent with the afterlife and his references to resurrection.
Among the Jews of Christ's time, the Pharisees believed in the resurrection of the dead into bodies, but the Sadducees, the sect of Temple priests, didn't believe in an afterlife of any form. In other words, the Sadducees followed the ancient Judaic tradition. They didn't believe in angels, the soul, or resurrection. Life was literally "dust to dust." While some Sadducees believed in a spiritual afterlife, they didn't agree with the Pharisees regarding "the rising." The view of the Pharisees was similar to that of the Egyptians, that people would rise from the dead with their bodies. This idea can be traced directly to Isaiah in the verse cited above. It is important to note that, especially for the Pharisees, this resurrection was connected to the coming of the Messiah, which, for them, was the end-time like Christians see the Last Judgment.
Jesus's first message here is, that, though Christ taught that there was a "rising" as taught by the Pharisees, it was not what the Pharisees taught, that is, something that happened at some end time when the Messiah comes.