1)
Why did Hashem raise His hand?
1.
Rashi (from Sifri 330): This was for a Shevu'ah. Radak - this is like "Ki Esa El Shamayim Yadi" (Devarim 32:40). "Lahem" is about them.
2.
Malbim: Corresponding to their two sins, He swore to give to them two punishments. (a) He will kill them in the Midbar; they themselves will not enter the coveted land that they despised. Also refer to 106:27:1:1.
2)
What is the meaning of "Lehapil Osam ba'Midbar"?
1.
Radak: He will kill all the complainers, who were 20 years old and above 1 - "ba'Midbar ha'Zeh Yiplu Figreichem" (Bamidbar 14:29).
1
Perhaps they complained, for they are the ones who fight in the army. (PF)