It has been seen that black holes throw out jets of high speed particles at almost the speed of light from it's poles along it's spin axis. Since a black hole has so strong a gravitational field inside the Schwartzschild radius that nothing (not even photons) can escape, how come these material particles escape and what force acclerates them to such high speed?
You are correct that high speed particles are accelerated away from the black hole in jets along the black hole's rotation axis. This jet phenomenon manifests itself on the scale of supermassive black holes as Active Galactic Nuclei and on the stellar black hole scale as "microquasars". Likely the physics is similar in both cases even though the size scales are very different. The short answer to your question is the particles that form Astronomers still study the details of jet formation and acceleration, but the prevailing idea is its due to strong magnetic fields. Imagine an accretion disk around a black hole. Because of the high temperatures in the disk, the gas is ionized and forms a plasma. There are also likely strong magnetic fields that thread through the disk, and because of the laws of electromagnetism, are essentially "frozen into" the accretion disk. That means as the particles spiral into the black hole they drag the magnetic fields causing them to twist. Eventually the magnetic fields will become so strong that instead of the particles dragging the magnetic field accelerates the particles toward the poles. They then go flying out in the form of jets. It turns out that jets are a common theme in astrophysics. In addition to being seen around black holes, they're also seen around star forming regions. Astrophysicists think that jets serve an important function of getting rid of angular momentum in a system. Otherwise the accretion disk or protostar would fly apart because conservation of angular momentum would require it to spin very rapidly.