As mentioned by curious, the induced sequence is in general not exact on the left, though it is exact at the other two places. But as both curious and Soarer point out, the image of the map on the left is $aM$ under the isomorphism $A\otimes_A M \cong M$, and then exactness at the other two places gives you the isomorphism that you want.
is there any other way to think about the tensor product besides been the quotient of a free module by a set of generators?
Yes! In fact, there is a much more useful, if more abstract, way to think of tensor products. Let $A$ be a commutative ring, and let $M,N,Z$ be $A$-modules. (Analogous statements hold over noncommutative rings, but the formulation is not quite as clean.) An $A$-bilinear map $b:M\times N\to Z$ is a map of sets such that for each $m\in M$ and each $n\in N$, the maps $b(m,-):N\to Z$ and $b(-,n):M\to Z$ are $A$-linear.
Bilinear maps appear all over the place; you can think of them as "generalized multiplications" -- for any $A$-algebra $R$, its multiplication $R\times R\to R$ is $A$-bilinear. Note that a bilinear map $M\times N\to Z$ is not $A$-linear with respect to the usual $A$-module structure on $M\times N$. That's too bad, because it's always nice to have things be linear. To "correct" this, the tensor product $M\otimes_A N$ is exactly the $A$-module such that bilinear maps $M\times N\to Z$ are the same thing as linear maps $M\otimes_A N\to Z$.
More precisely, $M\otimes_A N$ is characterized by the following universal property: if $Q$ is an $A$-module and $b:M\times N\to Q$ is an $A$-bilinear map, then there is a unique $A$-linear map $\tilde b:M\otimes_A N\to Q$ such that $\tilde b(m\otimes n) = b(m,n)$ for all $(m,n)\in M\times N$.
So, for example, if $R$ is an $A$-algebra, then the multiplication of $R$ is an $A$-linear map $R\otimes_A R\to R$. Indeed, this lets you phrase the axioms of an associative algebra completely in terms of linear maps, which is a useful thing to do if, for example, you want to talk about "algebra objects" in categories other than the category of $A$-modules.