It says in the Parasha of Taharoth, "Two live, clean birds shall be taken for the person being purified". Of the two birds taken, one was slaughtered and the other was sent away.
It says in the Talmud Yerushalmi, that Ribbi Shim'on Bar Yohai at first felt that man should have been created with two mouths and two tongues. One would be used for speaking holy matters, and the other for his worldly needs, because, he did not feel that it was appropriate for the same mouth that spoke holy matters to also speak profane. After seeing, however, how much Lashon Hara' human beings speak, he said how wonderful it is that Haqqadosh Barukh Hu (the Holy One Blessed be He) only created us with one mouth and one tongue. Because with only one look at how much wrong we do and could we imagine how terrible it would be had we been born with two?
Birds are a hint to the tongue because birds chirp constantly. The reason why at first two birds must be brought, is to hint that it would have been appropriate to have had two mouths and tongues, one for the service of G-d and the other for the profane. But, since G-d saw how much man would sin in this area He only gave us one. Therefore, one is slaughtered and the other is sent away, to represent that one was cancelled and we are left with just one mouth and tongue for both areas.
(See 'Od Yossef Hai Derashoth, Parashath Taharoth)
