Even though Moshe was told by G-d that his mission would succeed, he was also told that before the dawning of success, there would be repeated failures. Therefore, he understood that doubts would arise in the minds of the people and asked for a sign to reassure them that it was indeed G-d who sent him.
In the first of these signs, G-d told Moshe to throw his mateh, his staff, on the ground. Once on the ground, it turned into a serpent, whereupon Moshe fled from it. Then, upon G-d’s instruction, he took hold of the tail of the serpent and it turned back into a staff in his hand. How was this a sign to the people that the G-d of their forefathers sent Moshe?
On a simple level, it is a sign because any act that evidences the power to set aside the natural order is a sign of a Higher Power — a Power Who establishes and operates the laws of nature.
A more penetrating analysis, however, should also explain why G-d chose this particular sign instead of some other one. The staff — mateh — is a most natural symbol of man’s mastery over nature. It has a dual function, corresponding to the dual meaning of its root, nateh.One meaning is to lean or incline, and the other meaning is to stretch one’s hand over something.
Correspondingly, mateh denotes an extension of the hand, upon which man can lean for support as he stands on the ground, as in a cane, and an extension of man’s sphere of power, as in a scepter — a symbol of his authority.
This sign in Moshe’s hand will show the people that if G-d so desires, the thing on which a person leans for support and with which he wields authority can turn into the very opposite — a serpent which causes man to recoil.
The message to Moshe is: You have been sent by G-d, Who, if He so desires, can cause the very thing on which man relies for support, and which serves him as an instrument of his authority, to turn against him. Conversely, if He so desires, G-d can take a hostile form that is feared and shunned by man and place it into his hand as an accommodating support and tractable tool. He can make Pharaoh a slave and you are the ruler. He can turn Pharaoh’s staff into a whip for his own back. For nothing inherently supports or inhibits man — it is only G-d Who assigns such roles as staff and serpent.
Sources: Commentary Bereishet 4:2-5