Today - Tuesday - is Rosh Chodesh Sivan. It's the day the Yidden arrived at Mount Sinai, as it says in Parshat יִתְרוֹ (Exodus 19:1):
The Mitzva of Tefillin was given to the Yidden right after leaving Egypt - before they arrived at Har Sinai.
Men wear Tefillin every day during Shacharit (morning prayers) except for Shabbat and Yom Tov.
Rosh Chodesh has some aspects of a Yom Tov, yet work is permitted.
Tefillin are worn on Rosh Chodesh during Shacharit, Hallel and the Torah reading. They are removed before starting Mussaf.
Source: Kitzur Shulchan Aruch 10:19
Rosh Chodesh Siva 5775, 45th day of the Omer