Which one is grammatically correct and why? Please explain:

1) She gives permission for the students to use their cellphone.

2) She gives permission to the students to use their cellphone.

Question

Which one is grammatically correct and why? Please explain:

1) She gives permission for the students to use their cellphone.

2) She gives permission to the students to use their cellphone.

Answer

I would suggest:

She gave the students permission to use their cellphones. (for a one-time exception to a policy)

OR

She lets the students use their cellphones. (for an ongoing policy)

Answer

I prefer "to the students" because the verb "give" most often follows the pattern: "give + something + to + someone".

You can simplify the sentence by dropping one of your prepositions:
"She gives cellphone permission to the students"

Grammatically, there is no difference between your two sentences. Both are correct and it is difficult to detect any difference in meaning.