Rashi in Devarim 11 on the verse "to speak in them" quotes the Sifrei that when a child first begins to talk, his parents should speak to him in Hebrew and teach him Torah, and this will guarantee the child a long life. This same idea is found in the Tosefta in Chagigah which states that when a child knows how to talk, his parents should teach him Hebrew.

The clearest source which identifies learning Hebrew as a Mitzva is found in the Rambam's commentary on the Mishna in Avot (Chapter 2). The Mishna states that one must be careful regarding a minor Mitzva as one is with a major Mitzva. As an example of a minor Mitzvah, the Rambam cites studying and speaking Hebrew. The Rambam here clearly considers learning and speaking Hebrew to be a Mitzva, even though people think it is a minor Mitzva

The difficulty is that although the Rambam's view is clear in his Mishna Commentary, he does not codify this Mitzva of learning or speaking Hebrew in his Mishneh Torah, nor does such a Mitzva appear in the Shulchan Aruch. The Torah Temimah in Parshas Eikev wonders why the Poskim omitted any reference to the requirement to learn and speak Hebrew. However, Rabbi Moshe Feinstein in Iggros Moshe Even Haezer (Chapter 35), actually states clearly that there is a Mitzva to speak Hebrew. Although he states there is no prohibition to speak in any other language.

The Ramban in Parshas Ki Sissa writes that Hebrew is called Loshon Hakodesh precisely because it is the language used by the Holy Torah. The Yerushalmi in Shabbos states that one who speaks Loshon Hakodesh is guaranteed to be in Olam Haboh. The Korban Haeidah explains this Talmudic passage that speaking Hebrew leads to Spiritual Purity.

It is interesting to note that the Shulchan Aruch in O. C. (Chapter 307) rules that whereas it is not appropriate to read certain types of literature on Shabbos, but if they are written in Hebrew, they may be read on Shabbos. The Magen Avrohom explains that this is because the Hebrew language itself has Kedushah, and one can learn how to learn Torah simply by reading books and even letters written in Hebrew.

The Midrash in Vaykra Rabba states that even though the Jews were in 49 levels of impurity in Egypt, they were still worthy of Redemption because they continued to speak Hebrew. The Midrash brings other reasons why we merited Redemption, but we see that speaking Hebrew is your identity card as being a member of the Jewish People.