The determinant of block triangular matrix as product of determinants of diagonal blocks
Use a proof by induction on $m$:
$(a)$ It seems to me, then, that you have your base case $m = 2$: $P(2)$,
$\quad$ though it suffices, for a base case, to prove it's (trivially) true for $m = 1$.
$\quad$ But the process used in class for proving it's true for $m = 2$ will be helpful when making the
$\quad$ inductive step of the proof.
$(b)$ Your inductive hypothesis $P(k)$ would be to assume that this is true for $m = k$.
$\quad$ That is, assume the truth of: $$P(K):\quad \det \begin{bmatrix} A_1 &* &* &* &* &* \\ 0& A_2 &* &* &* &* \\ .& 0& A_3 &* &* &* \\ .& 0& 0 &... &* &* \\ .& 0& 0& & ... &* \\ 0& .& ...& 0&0 & A_k \end{bmatrix} = \prod_{i=1}^k \det A_i $$ $(c)$ Then, take the inductive step: you'll need to use the inductive hypothesis to prove that $P(k+1)$ is true: for $m = k + 1$... That is, assuming $P(k)$ is true, prove: $$ \det \begin{bmatrix} A_1 &* &* &* &* &| &* \\ 0& A_2 &* &* &* &| &* \\ .& 0& A_3 &* &* &| &* \\ .& 0& 0 &... &* &| &* \\ .& 0& 0& ...& A_k &| &* \\ \hline& & & &\\ 0& 0& ...& 0&0 &| & A_{k+1} \end{bmatrix} = \prod_{i=1}^{k+1} \det A_i $$
Note that to do this, you can partition the matrix into two block matrices on the diagonal,
$(1)$ one of which is triangular (block) matrix with $k$ sub-blocks $A_i$ for $1 < i < k$ along it's diagonal, for which the determinant you know from the inductive hypothesis (having assumed its truth), and
$(2)$ the other with one block on the diagonal which we call $A_{k+1}$.
Here is where you can use the proof used in class for $m = 2$
That is: $$\prod_{i=1}^{k+1} \det A_i = \left(\prod_{i=1}^{k} \det A_i\right)\cdot \det(A_{k+1})
$$
Then you will have shown that the determinant of a partitioned triangular matrix is product of the determinants of the block matrices on the diagonal.
Sketch of an alternative argument: the determinant of an upper triangular matrix is the product of its diagonal entries. You can do elementary row and column operations to the large matrix that make each $A_i$ upper triangular without changing any of the entries in $A_j$ for $i \neq j$ (taking account of any row swaps you may have to perform). Once you have done so, the whole matrix will be upper triangular, so its determinant is the product of the diagonal entries (and then correct the sign according to the parity of the row swaps you have performed).
I don't think this is what your instructor wanted since it doesn't use the 2 by 2 case, but it's how I think about this fact.