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Since thrusters have no matter to push off of, how can a spacecraft generate thrust in the vacuum of space?

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By conservation of momentum.

When the thruster expels high velocity gas in one direction, that gas has momentum.

Since there is no external force acting on the system, the total momentum of the system (thruster and expelled gas) cannot change.

Thus, the thruster must acquire an opposing momentum such that the total momentum is unchanged.

See the Wiki article Tsiolkovsky rocket equation

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So the spacecraft pushes off of the expelled gas? – dongle26 Oct 5 '12 at 16:54
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Consider two masses at rest connected by a compressed spring. If the spring is suddenly allowed to decompress, the two masses fly away from each other but the total momentum is still zero. They "pushed off of each other". – Alfred Centauri Oct 5 '12 at 16:56
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Yes, the spacecraft pushes off of the expelled gas. That's correct.

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