Your vet is talking about a Saddle Thrombus:
This blood clot can leave the heart in the large blood vessels exiting the heart, but then get stuck in many of the smaller blood vessels leaving those major arteries. When this happens, it can cut off blood supply to a limb. Suddenly the cat limps on that limb and is very painful.
Think of wrapping an elastic band tightly around your finger. Pretty quickly it becomes painful and it is hard to use the finger. This is essentially the same thing that happens when a thrombus blocks blood supply to a cat's limb.
On exam, these kitties present with pain and limping. If the clot has gone to the right front leg (the first "exit" from the major arteries) that foot will be colder than the others and the nail beds will be blue. The blood clot may lodge in the caudal vena cava, which is a major blood vessel leading to both back legs. In this case, the cat will not be able to use the back legs but will drag them around and will be in severe pain. If this happened some time ago, the limbs may be so numb by now that there is no pain from them anymore. However, the pads will be notably cool, the nail beds will be blue and the cat will not have pulses on the back legs. This carries a grave prognosis.
Early treatment involves drugs to stop more clots from forming - the choices are aspirin or coumadin (rodenticide, rat poison, warfarin). Unfortunately, if there is a major clot to both back legs, treatment may not be possible.
Here is more about thrombus formation in cats:
And more about Hypertrophic Cardiomyopathy which is the most common heart problem in cats, and the most common reason they develop blood clots:
I hope he will be treatable. Please let me know how he is doing.