You can read Ernest L. Martin's The Star of Bethlehem: The Star That Astonished the World for free here-

https://www.askelm.com/star/

Here's a link to the specific chapter where Martin describes how he derived the exact date-

https://www.askelm.com/star/star006.htm

You can read the rest of it for further investigations and proofs, but here's the single paragraph explaining the date itself-

The Precise Position of the Moon is Important

 Now note this point. Since the feet of Virgo the Virgin represent the last 7 degrees of the constellation (in the time of Jesus this would have been between about 180 and 187 degrees along the ecliptic), the Moon has to be positioned somewhere under that 7 degree arc to satisfy the description of Revelation 12. But the Moon also has to be in that exact location when the Sun is mid-bodied to Virgo. In the year 3 B.C.E., these two factors came to precise agreement for about an hour and a half, as observed from Palestine or Patmos, in the twilight period of September 11th The relationship began about 6:15 p.m. (sunset), and lasted until around 7:45 p.m. (moonset). This is the only day in the whole year that the astronomical phenomenon described in the twelfth chapter of Revelation could take place.

 This also shows one other important point. The Moon was in crescent phase. It was a New Moon day, the start of a new lunar month. (See plates one and two below which show early depictions of the celestial scene of Revelation 12:1–3 and how the Moon is shown to be in its crescent phase.)