Where did Moshe place the broken Luchos?
Rashi (in B'rachos, 8b) and Ramban: Hashem instructed him to place them in the Aron together with the second Luchos. 1
Refer also to 10:1:3:1-3.
After Betzalel built the Aron for the Mishkan, the Beraisa Meleches ha'Mishkan (Perek 6) and R. Yehudah ben Lakish (Yerushalmi Shekalim 6:1) hold that the Torah and the broken Luchos were in separate Aronos. R. Bechayei - the latter (whole) Luchos were in the Aron in the Ohel Mo'ed (with the Torah), like Bamidbar 10:16 says. The broken Luchos were in the Aron that Moshe made. Tosefta Sotah 7:18 and Rabanan in the Yerushalmi hold that the whole and broken Luchos were in the same Aron. Da'as Zekenim (1), Hadar Zekenim (1), Rosh - that this was only in the Beis ha'Mikdash; before this, they were in separate Aronos.
Why did Hashem see fit to add the words "Asher Shibarta"?
Targum Yonasan: They are the acronym of "Yeyasher (Kochacha) she'Shibarta"
See the last Rashi in the Torah.
Refer to Shemos 34:28:1:1,2 and the notes there.