In Act 1, Scene 2 of Julius Caesar, Brutus is questioning Casca about the crowd noises he and Cassius had been hearing offstage. Casca tells Brutus that Caesar "swooned and fell down." Cassius seems surprised. He asks Casca to explain and Casca elaborates:

He fell down in the marketplace, and foamed at mouth, and was speechless.

Brutus is not surprised to hear this. He comments:

'Tis very like: he hath the falling sickness.

The symptoms Casca describes are those of epilepsy. Obviously they did not have that term for it in ancient times. Caesar had a sickly childhood but an exceptionally strong will power. He determined that he would not allow his physical problems to prevent him from becoming a great soldier and a great man. He forced himself to endure all kinds of physical exertions and privations, as described by Plutarch in his "Life of Caesar." Cassius earlier in Act I, Scene 2 tells Brutus how Caesar once challenged him to dive into the Tiber River and swim across it during a storm while the water was turbulent. This is illustrative of the kinds of physical challenges Caesar imposed on himself. It would seem that Caesar's compensation for his infirmities was partly responsible for his great accomplishments as a general, which led to his ascendancy in Roman politics. 

The "falling sickness" was a name for epilepsy. It means that he would sometimes have seizures, lose control of his body, and fall to the ground. Depending on the type of epilepsy, some bodily function might be lost (twitching, convulsions, etc.) It was associated with possession at some points in history, and so with communication with spirits.

Caesar was epileptic - prone to seizures. From there, however, Brutus and Cassius turn the falling sickness into a metaphor to stand for all Romans who are too weak to stand up to Ceasar's tyranny.