Good question. Rabbi Moshe ben Nachman (Ramban, Nachmanides) explains this seemingly odd practice in his commentary on the Torah.
The Torah refers to the months by their chronology from the time of the Exodus from the slavery in Egypt. The "first month" is when they left Egypt, and then the "second month" etc. through the months of the year. This teaches and reminds us of the great miracles and lovingkindness that God showed the Jewish nation by taking them from exile and beginning their national purpose of receiving the Torah and entering Israel.
In a similar way the present names we use for the months (Nissan, Iyar, Sivan, etc.) were imported into our Hebrew usage when we returned to the Israel from the Babylonian exile after the fall of the Babylonian empire. This deepens our remembering the end of the exile. When we use the Babylonian names for the months we are further reminded of God’s redemptive presence in history. It was not just one time that God did this for us back in the time of Egypt. The second exile, this time to Babylonia, came to end as well through the special all-knowing, all-powerful - and always present - actions of God.
