Why is shatnez (a garment made of a mixture of wool and linen, which is forbidden) muter (permissible) by tzitzis?
In short: The Torah says in Devarim 22: 11 - 12:
You shall not wear a mixture of wool and linen together. You shall make yourself twisted threads, on the four corners of your garment with which you cover yourself.
We learn from this juxtaposition of pesukim that one can even make Tzitzis in a case of Shaatnez.
See Menachos 39b-40a. Shaatnez is allowed for tzitzit by Torah law, at least in some cases (in the presence of techeiles, it's complicated, see 40a Tosefos d"h כיון), for the reason given by yydl from the gemara in Yevamos 4a. However, the gemara there in Menachos says that you are not allowed to do it in practice, a Rabbinical enactment for one of several reasons given there - [we're not sure it's real techeiles dye, one might do it with a thread that wasn't attached with the proper intention, one might come to wear it at night when there is no mitzvah] - where the mitzvah is not fulfilled, in which case there would again be a prohibition of Shaatnez.