Given a vector space $V$ over some field $F$, we'd like to consider rotations or scalings and whatnot, which come down to defining functions $f: V \rightarrow V$ on our space which respect certain aspects of $V$, i.e. linear transformations. This idea generalizes to other algebraic or geometric objects: we want to look at functions on those objects that behave nicely with respect the the objects' structure. So given such a transformation $f: V \rightarrow V$, we would require that $f$ respect a basis change.
Consider complex lattices $\Lambda = \omega_{1}\mathbb{Z} \oplus \omega_{2}\mathbb{Z}$, with $\{\omega_{1}, \omega_{2}\}$ being a basis for $\mathbb{C}$ over $\mathbb{R}$. These correspond to complex elliptic curves $\mathbb{C}/\Lambda$ via the Weierstrass $\wp$-function (see the Wiki article). Further we can identify each such $\Lambda$ with $\Lambda_{\tau} = \tau\mathbb{Z}\oplus \mathbb{Z}$, where $\tau$ is the quotient of either $\omega_{1}/\omega_{2}$ or $\omega_{2}/\omega_{1}$ according to whichever lies in the upper half plane $\mathbb{H}$. Thus, elliptic curves can essentially be identified with elements of the upper half plane $\mathbb{H}$. The catch is that two such curves $\mathbb{C}/\Lambda_{\tau}$ and $\mathbb{C}/\Lambda_{\tau'}$ are "equivalent" (isogenous) when there is some $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_{2}(\mathbb{Z})$ such that $\gamma(\tau) = \tau'$.
Following the vector space analogy, we'd like to look at functions on the elliptic curves $\mathbb{C}/\Lambda_{\tau} \rightarrow \mathbb{C}/\Lambda_{\tau}$, i.e. functions $\mathbb{H} \rightarrow \mathbb{H}$. Thus, such a function should respect the equivalence (isogeny) given above: $f(\gamma(\tau)) = f(\tau)$. This should mirror the requirement that linear transformations respect basis change of $V$. Given the model of linear transformations from $V$ to $V$ (from a vector space to itself, or from a vector space to another vector space, respecting angles and scalings), why not consider "transformations" from one curve $\tau$ to another $\tau'$? Well, these seem to be the $\gamma \in SL_{2}(\mathbb{Z})$ above, right?
If this is true, we somewhat know what the (interesting) maps $\mathbb{H} \rightarrow \mathbb{H}$ are. So why do we consider (as modular forms) the maps $\mathbb{H}\rightarrow \mathbb{C}$ which respect isogeny? That is, what is the motivation behind considering functions on classes of curves $\{\mathbb{C}/\Lambda_{\tau}: \tau \in \mathbb{H}\}/\{\tau \equiv \gamma\tau'\}$?
$\Bbb{C/(Z+iZ)}$ is isogeneous but not isomorphic to $\Bbb{C/(Z+2iZ)}$ (the later doesn't have the $z\to iz$ automorphism).
$f\in M_k(SL_2(\Bbb{Z}))$ iff $f(\tau)=F(\Bbb{Z}+\tau \Bbb{Z})$ with $\tilde{F}(u,v)=F(u\Bbb{Z}+v\Bbb{Z})$ analytic on $\{ (u,v)\in \Bbb{(C^*)^2}, u/v\not\in \Bbb{R}\}$ and $F(w \Lambda) = w^k F(\Lambda)$.
and $F(\Bbb{Z}+v\Bbb{Z})$ is bounded when $\Im(v)\to +\infty$.
The last condition is to make $M_k(SL_2(\Bbb{Z}))$ finite dimensional (or analytic at $i\infty$), without it then $M_4(SL_2(\Bbb{Z}))$ would contain $j^n G_4$ for all $n$.
Note that $\tilde{F}(u,v)$ is $GL_2(\Bbb{Z})$ invariant.
There is also an algebraic definition: $f\in M_k(SL_2(\Bbb{Z}))$ iff $f$ has no pole on $\Bbb{H}\cup i\infty$ and $f / (j')^{k/2}\in \Bbb{C}(j)$. $j'$ is better understood as the meromorphic 1-form $dj=j'(z)dz$ and $(j')^{k/2}$ as a tensor product $dj\otimes_{\Bbb{C}(j)}\ldots \otimes_{\Bbb{C}(j)}dj$.