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 תְּפִלִּין was given to the Yidden right after leaving Egypt - before they arrived at Har Sinai.
Men wear תְּפִלִּין 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, הַלֵּל and קְרִיאַת הַתּוֹרָה. They are removed before starting מוּסָף.
Source: Kitzur Shulchan Aruch 10:19
Rosh Chodesh Sivan 5776, 45th day of the Omer