Jesus uses this phrase in Matthew 22:32,
In all of them the contrast is made between a God of living beings not dead ones. This idea has deep roots in the philosophy of the area.
The ancient world had a very clear concept of a "god of the dead." From the earliest Egyptian religions from the time of Moses to the Roman religions of Christ's era, there was a clear model for the afterlife. Souls, referred to as "the dead," traveled to the underworld where they were judged by the god of the dead (Anubis in Egypt, Pluto in Rome, Hades of the Greeks). The "dead," no matter how they were judged, lived a type of half-life, without a body (though among the Romans, some thought that the most heroic dead could achieve a godlike status in the underworld and could affect our world). The Egyptian religion was built around the idea that spirit of the dead would joined by resurrection of the body at some future date, but that idea was not widely shared in the ancient world.
For Christ, the statement that God is not "a god of dying" may speak directly against the existence of an afterlife that is some form of holding cell where spirits kill time until the resurrection but this would be cleared if he had used an article to say "the dead" instead of the adjective "dying."
Christ may be saying that the Patriarchs are living and have always been living, at least from God's point of view. He makes it clear that he is not talking about life of the soul, but a "resurrected" life. In Mat 8:11, Christ describes them as having their own table in the heavens where Gentiles will join them. This "beyond earth" (see this article about the meaning of the word "heaven") elsewhere.
Here, as in many other places, Christ is dealing with our faulty perceptions of time. As we now know, time is a dimension of our physical universe, just like height, depth, and breadth. Christ is saying that God, angels, and our own resurrection exist outside time, that is, outside of our current framework of existence.
From God's point of view, Abraham, Isaac, and Jacob are living in every sense. Their life in what we consider the past exists now for God, but their resurrected lives also exist now. Though we cannot conceive of a timeless state or a state which has direct access NOW to all of time, this is a limitation of our current state, not a limitation on God, the angels, or our future state.
Because we cannot think outside of time, we try to force both God and the afterlife to conform with our perspective, which exists within time. So, we conceive that, after we die, time passes until the Last Judgment. We have to spend that time in a spiritual heaven, which is different than the final paradise, or we have invent a concept call "soul sleep," which assumes our spirit "sleeps" until that final Judgment Day.
For Christ and God, time does not pose a problem. Creation through all of time exists as a whole. Any part of it is directly accessible. There is no real "before" or "after" except as different points, like different points in space. God didn't walk with Adam and Eve in paradise before or after Christ's death and resurrection in the Roman ear. God is walking in Eden. Christ is walking on earth. The Judgment Day is happening now. We are being born now. We are dying now.