The approvals are letters from two Rabbis and the sons of the Alter Rebbe, and they say two important things:
1) The Alter Rebbe was worried that someone might change his words and make it seem as if he were saying something contrary to the Torah, Therefore, he had the book printed and asked these important rabbis to write expressing that they had read the book and that they liked everything it said.
2) That nobody can print the book for five years or cause harm to the printing company that worked on the book, because there were many people who were not happy with the printing of Tania.
One of these rabbis who wrote endorsements was Rabbi Zusia of Anipoli, and he said, "We will go out to meet the..." Messiah with Tania! Messiah He is a person we are waiting for to come and take us all to the Land of Israel and reveal God's presence everywhere.
Another rabbi, Rabbi Levi Yitzchak of Berdichev, when he saw the Tanya, said that it is impressive how the Alter Rebbe was able to put such a great God into such a small book…
The approval of the Alter Rebbe's sons also adds that there is a section in the Tanya where they will print a series of letters from the Alter Rebbe and that no one can copy the Tanya without permission.