The stars symbolized the one community of the galaxy. It is mentioned in the article on Heir to the Empire Sourcebook (pg. 11):
The Provisonal Council of the New Republic government has adopted a
seal which incorporates the symbol of the Alliance with a pattern
signifying the member worlds of the galactic community.
The blue crest has been placed upon a circle of stars, signifying the
galactic community. The New Republic is more than the government and
the military, it is the people. The field of stars represents those
people - whether they be Human, Wookiee, or some other galactic
The specific number of the stars is not explained in the article. However, the Declaration of a New Republic was signed by eight leaders of the Alliance of Free Planets, who later formed the New Republic Provisional Council. Only the total of nine councilors and their worlds are introduced, but with an additional mention:
Other members of the Council have included unknown representatives of
the Ithorian and Sedrian species.
Since, according to Wookieepedia, the Declaration of a New Republic is quite similar to the United States Constitution, and the Declaration of Rebellion is like the United States Declaration of Independence, assumably the creators of the media followed the same line here, too. The number of fifteen stars most likely represented the original founding worlds, just like the stripes on the US flag represent the thirteen colonies that rebelled against the United Kingdom.