Having informed us in Pasuk 1 that Moshe was instructed to count Yisrael on the first of Iyar, why does the Torah repeat the date here?
Ramban: To teach us Moshe's Z'rizus (zealousness in performing a Mitzvah) - how on the day that Hashem instructed him to count the people, he gathered the princes and the people, and began to count them. 1
What is the meaning of the words "Vayisyaldu al Mishpechosam"?
Rashi: It means that they came with their birth-certificates and witnesses of birth. 1
Ramban: It means that the offspring were counted together with their fathers' families. 2
Targum Onkelos and Targum Yonasan: It means that they established their genealogy.
See Ramban's objection to this explanation.
Ramban: They all came with their half-Shekalim - all except for the Eirev Rav - and declared before Moshe and the princes 'I, P'loni, the sin of P'loni, from family P'loni, from the tribe of Reuven (on such and such a date). Moshe would then place the Shekalim of each tribe in a separate location, so that the total of each tribe, as well as the sum total of all the tribes would become known. See Ramban as to why the Torah omits the phrases "Vayisyaldu al Mishpechosam" and "be'Mispar Sheimos le'Gulgelosan" from the census in Pinchas.
Why was this necessary?
Seforno: Because they were about to begin Cana'an, and when a Jewish army goes to war, the pure genealogy of the soldiers is essential. 1
Seforno: See Kidushin, 76a, and as the Gemara states there, 70b 'The Shechinah rests exclusively on the families of Yisrael that have a pure pedigree'.
Having written "be'Mispar Sheimos Kol Zachar le'Gulgelosam" in Pasuk 2, why does the Torah repeat it here?
Ramban (citing Midrash Rabah): It teaches us that Hashem instructed Moshe to count every single Jew in the most dignified manner. He was not to ask the head of the family how many members were in the family and how children he had, but each member was to pass before him and to be counted individually.