M-theory compactified on a 2-torus is the same as M-theory compactified on a circle and then compactified on another circle because $T^2=S^1\times S^1$.
M-theory compactified on a circle is type IIA string theory with $g_s$ being an increasing power of the radius of the compactified dimension. And if type IIA is compactified on a circle of a small radius, we get type IIB string theory via T-duality. When we connect the M-theory/IIA duality and the IIA/IIB T-duality, we get the $F=MA$ relationship between M-theory and type IIB you mentioned.
One may avoid the type IIA intermediate step, too. M-theory on a two-torus is naively a 9-dimensional theory (the supergravity approximation would lead us to this belief). However, M-theory contains M2-branes, two-dimensional objects, and both of their spatial dimensions may be wrapped on the 2-torus. This produces point-like objects in the remaining 9 large dimensions. These objects are light when $A\to 0$ and they also have bound states of $N$ objects. So one obtains a continuum of new states in the $A\to 0$ limit and they may be reinterpreted as the momentum modes with respect to a new, "emergent", 10th spacetime dimension of the resulting type IIB string theory.