Q. Why do they keep Rosh HaShanah for two days in Israel, although only one day is observed there on the other festivals?
A. Although calendrical doubts led the Diaspora to double up and keep two days of each yom-yov, a different consideration ensured that even in Israel there would be two days of Rosh HaShanah. This was the concept that Rosh HaShanah is yoma arichta, one long day, lasting in fact 48 hours.
A reason is that this is Yom HaDin, the day of judgement, when our lives are in the balance. The halachic rule was that in cases involving life and death, a Jewish court did not complete the proceedings in one day. By leaving the completion of the case to the next day there was the possibility of finding grounds for acquittal.
We are all on trial on Rosh HaShanah. This is why people should not give themselves a dispensation to ignore the second day and not attend the service.