What did Moshe mean when he said to the Levi'im "Mil'u Yedchem ... "?
Seforno: He meant that they should acquire perfection for their hands today to be ready to serve Hashem in His Mikdash. 3
Rashbam: He meant that each one of them has consecrated his hands with a Korban on behalf of Hashem. 4
Targum Onkelos and Targum Yonasan: He meant that they should bing Korbanos today to atone for the blood that they spilt!. 5
Refer to 28:41:2:1*.
To take over from the Bechoros, who had been the priests until now.
By stretching out their hands to kill their own sons and brothers for the sake of Hashem. See also Bamidbar 32:12 (Rashbam).
What reward did the Levi'im receive for this act?
Da'as Zekenim and Hadar Zekenim (in Pasuk 26): Moshe blessed them 1 in Parshas ve'Zos ha'Brachah, even though Ya'akov had not blessed Levi. 2
Mishnas R. Aharon (4 p.21): They merited "Yoru Mishpatecha le'Yisrael" because at a time of slackening, they did not bow to the Eigel; but the contrary, they answered Moshe's call "Mi la'Hashem Elai!"
Da'as Zekenim and Hadar Zekenim: Likewise, Moshe blessed Reuven (even though Ya'akov did not), because they fulfilled the Mitzvah to cross the Yarden and fight - He had Ru'ach ha'Kodesh to know that they would do so. Shimon, on the other hand was not blessed by either Ya'akov or Moshe (PF)