What does the /= operator in C# do and when is it used?
It's divide-and-assign.  x /= n is logically equivalent to x = x / n.
It is similar to +=, -= or *=. It's a shortcut for a mathematical division operation with an assignment.  Instead of doing
x = x / 10;
You can get the same result by doing
x /= 10;
It assigns the result to the original variable after the operation has taken place.
A division and an assignment:
a /= b;
is the same as
a = (a / b);
Its simply a combination of the two operators into one.
In the following example:
double value = 10;
value /= 2;
Value will have a final value of 5.
The =/ operator divides the variable by the operand (in this case, 2) and stores the result back in the variable.
In most languages inspired by C, the answer is: divide and assign. That is:
a /= b;
is a short-hand for:
a = a / b;
The LHS (a in my example) is evaluated once.  This matters when the LHS is complex, such as an element from an array of structures:
x[i].pqr /= 3;