Why do long division remainders give conversion from base 10?

I learned that you can convert base 10 numbers to other bases, like binary, with long division.

I can do this, but I don't understand why this works. I can only understand that the first remainder of 1 gives a 1 in the 1's place (in binary). I can't understand why a remainder of 1 would give a 2, or a 4, or... etc.

$$N=a_nb^n+a_{n-1}b^{n-1}+...+a_1b^1+a_0b^0$$ $$=b(a_nb^{n-1}+a_{n-1}b^{n-2}+...+a_1)+a_0$$
Where $a_0$ is your first remainder. To get the second remainder you have to divide the expression by $b_1$ which would give you $a_1$, and so on.