I'm not sure if I wrote the Tex code right so here is a screenshot:
(the text says: "Remainder of the power:(...) with (...). If (...), the remainder is (...), otherwise we divide (...) by d and that way we get the remainder.")
I understand most of the notation except that asterisk in $q_*d$ and the $r\binom{n}{1}$. I understand what a combination is and I can see that they are here because Newton's binom is applied somehow, I just don't see how. I am not even sure if that is a combination because the notation isn't even quite the same.
I also don't understand why the remainder is divided by the divisor (I get it it's for the sake of simplification but why the divisor specifically?)
The $q_1$, $q_*$ and $r_1$ are just some numbers, and $r_1^n$ is $r_1$ to the power $n$. The subscripts $1$ and $*$ don't have any particular meaning except to show that these are different numbers that play similar roles in the text.
We begin from a number $a$, which has a remainder of $r_1$ when divided by $d$; then $a = q_1 d + r_1$ for some $q_1$ and $r_1 < d$. We want to compute the remainder of $a^n$ when it's divided by $d$. Of course, $a^n = (q_1 d + r_1)^n$. Now we use the binomial formula on the second expression: $$ (q_1 d + r_1)^n = (q_1 d)^n + n (q_1 d)^{n-1} r_1 + \cdots + \binom{n}{k} (q_1 d)^{n-k} (r_1)^k + \cdots + (r_1)^n. $$ Notice that every term except the rightmost one is divisible by $d$, which means that their sum is also divisible by $d$, and hence it has the form $q_* d$ for some number $q_*$. So we have $a^n = (q_1 d + r_1)^n = q_* d + r_1^n$. The first term $q_* d$ doesn't contribute to the remainder because it's divisible by $d$, so we can forget about it and just divide $r_1^n$ by $d$.