Early representations of giants actually looked normal and human. Consider this representation of Alcyoneus, the king of the giants:
(source: theoi.com) (http://www.theoi.com/Gallery/L1.2.html)
It was in fact only in the Hellenistic era did giants acquire feet of snakes.1(Pausanias' Description of Greece, translated with a commentary by J. G. Frazer, vol. 6 1898.) Why exactly they were chosen to be represented this way is unknown, but it makes sense, considering that the giants are the offspring of Gaia (the earth), and serpents in Mediterranean religion (and elsewhere) are—for very obvious reasons—intimately connected with land and with Gaia herself. It was probably just to symbolically show their origins.
1: Pausanias' Description of Greece, translated with a commentary by J. G. Frazer, vol. 6 1898.