Why does Roch Chodesh sometimes fall over two days, e.g., 30th of Nisan and 1st of Iyar?

Answer

In the times of the Holy Temple, the Sanhedrin actually determined the date of the new month, based on the citing of the moon. This is a special power that God gave the Jewish people to control time, and is irrespective of any scientific knowledge.

Be that as it may, information as to the proper date often did not reach the Diaspora communities until many days later, and they would celebrate two days of Rosh Chodesh out of doubt.

In the 5th century BCE, when Jewish unity was threatened by the exile from Israel, the patriarch Hillel II set a perpetual calendar, which included certain months that retained the status of "doubt" regarding when they would begin.

He did this even though he had full awareness of the precise dates of all the months. The Talmud (Rosh Hashana 25a) already had pinpointed the length of the lunar month as 29.53059 days. It wasn't until the atomic age that NASA scientists -- using satellites, hairline telescopes, laser beams and super-computers -- were able to calculate the lunar month as 29.530588.

So why was a second day added? In order to keep alive the concept of the Sanhedrin sanctifying the months, as the Torah prescribes.