Explain why you can use subtraction to solve a division problem

If I understand your question clearly, the reason why you can use subtraction in a division problem is that, in order for you to find out what a remainder or mixed number is, you need to subtract. 

Example: Tamira has 60 bags of cookies to give out at her local community center, each person receives 5 cookies in a bag, how many people got a bag of cookies? Is there a remainder left?

So, after reading that problem, you already know you need to divide.

 5 )‾‾60‾‾‾‾   

How many times can 5 go into 6? 1 time. So, you put the 1 over the 6.

      1
5 )‾60‾‾‾‾‾‾
   -  5 ↓ 
 
     1 0

How many times can 5 go into 10? 2 times.

     12
5)‾‾60‾‾‾
     -5↓

      10
     -10
 
        0

So, Tamira gave 12 people cookies with no remainder. And for when you need to find out the mixed number, it's the same process, but you go in the order of: quotient, remainder, dividend.

I hope I helped :)