Maharal: Sivan: The Significance of the 3rd
The reason that the Holy One, Blessed is He, waited until the third month, Sivan, to give the Torah, was in order for the Children of Israel to completely separate themselves from the past. “Pirud,” or separation, happens
in the third: The First connects to that which came before. The Third is the introduction of the new, and the second separates the first and third, allowing for Pirud, breaking away from the past.
It is therefore appropriate that we use the days between Rosh Chodesh Sivan and Shavuot to separate ourselves from the parts of our past that can limit our joy in Torah and our connection with it.