The division algorithm for polynomials in $F[x]$ The proof in here.
I am not really understanding how we know we should subtract $\frac{b_t}{a_m}x^{t-m}$. I understand everything else about the proof however.
 A: The key idea is to scale the divisor by $\,\color{#c00}{ax^k}$ so  it has the same leading term as the dividend, thus subtracting it from the divided will kill its leading term. Iterating we can keep killing the leading term, decreasing the degree till the whittled dividend has smaller degree than the divisor.
More explicitly, if the leading coefficient of the divisor $= 1$ (or is invertible), and the dividend has degree $\ge$ the divisor, then we can  $\rm\color{#c00}{scale}$ the divisor so that it has the same degree and leading coef as the dividend, then subtract it from the dividend, thereby canceling the leading term of the dividend; then recursively apply this process to the remaining part of the dividend, which has smaller degree (since we killed the leading term of the dividend), viz.
$$  (\overbrace{a x^{\large k+n} + f}^{\rm\large dividend}) - \color{#c00}{a x^{\large k}} (\overbrace{x^{\large n} + g}^{\rm\large divisor})\ =\ f-ax^{\large k}g$$
$$\ \Rightarrow\ \ \dfrac{a x^{\large k+n}+f}{x^{\large n}+g}\, =\ \color{#c00}{a x^{\large k}} +\!\!\! \underbrace{\dfrac{f-ax^{\large k} g}{x^{\large n} + g}}_{\large\rm recurse\ on\ this}$$
where the second equation arises from the first by dividing through by $\,x^n + g.\,$ The long division algorithm for polynomials is simply a convenient tabular arrangement of the process obtained by iterating this descent process till one reaches a dividend having smaller degree than the divisor.
Remark $\ $ If one seeks a deeper understanding one can view this as a special case of  multivariate polynomial division algorithms, such as the  Gröbner basis algorithm. One gains further insight from this more general perspective on the descent process, e.g. in terms of monomial orderings.
