I doubt that anyone, including the drafters of the legislation, could explain what they mean by "a Jewish state". However pledging allegiance to Israel as the national homeland of the Jewish people has some substantive meaning--that the purpose of Israel is always to be the place where any Jew who chooses can come home to. It would take a perverted mind to invent that this means that non-Jewish citizens of Israel have less rights than Jewish citizens. All of this would be so more easily defined if Israel would adopt a written constitution.
Seven Libyan soldiers killed in clashes with ISIS (Reuters)
from the article: No Knesset majority for loyalty oath for Jews and non-Jews