If $M_k$ denotes the space of weight $k$ modular forms with coefficients in $\mathbb Z$, then there is an embedding $M_k \hookrightarrow \mathbb Z[[q]]$ given by taking $q$-expansion. This induces a map $\bigoplus_k M_k \rightarrow \mathbb Z[[q]]$ which is again an embedding (although not longer quite obviously so).
If we let $\widetilde{M}_k$ denote the image of $M_k$ in $\mathbb F_{\ell}[[q]]$, then there is similarly an induced map $\bigoplus_k \widetilde{M}_k \rightarrow \mathbb F_{\ell}[[q]],$ but this is no longer an embedding. If we denote its image by $\widetilde{M}$, then this is the ring of mod $\ell$-modular forms. It is the sum of the various $\widetilde{M}_k$s, but is not their direct sum.
Assuming $\ell \geq 5$, its kernel is generated by $E_{\ell} - 1$. Note that this is not a homogenous element of the source. Thus the image is not naturally graded. However, any graded ring is also naturally filtered --- in our case we filter the direct sum by the subobjects $\bigoplus_{i = 0}^k \widetilde{M}_i$ --- and the image of a filtered ring is naturally filtered --- in our case we define $\widetilde{M}_{\leq k}$ to be the image of $\bigoplus_{i = 0}^k \widetilde{M}_i$ in $\mathbb F_{\ell}[[q]]$.
Then $\widetilde{M}$ is the union of the $\widetilde{M}_{\leq k}$. We say that an element of $\widetilde{M}$ has filtration $k$ if it lies in $\widetilde{M}_{\leq k}$, but not in $\widetilde{M}_{\leq k-1}$.
So, regarding motivation: it is the what replaces the notion of weight for an element of $\widetilde{M}$. In short, if we are handed a $q$-expansion in $\mathbb F_{\ell}[[q]]$ and told that it is the $q$-expansion of a modular form mod $\ell$, the weight is not intrinsically determined by the $q$-expansion (unlike in the case with char. $0$ coeffients), since e.g. the $q$-expansion $1$ is the $q$-expansion of the wt. $0$ modular form $1$, the weight $\ell-1$ modular forms $E_{\ell -1} $, and more generally the weight $(\ell-1)i$ module forms $E_{\ell -1}^i$ for any $i$. But the filtration of the $q$-expansion is well-defined. When you sort it out, it is essentially the minimal weight of a modular form having that given $q$-expansion.
As for our more specific question: since the kernel of the $q$-expansion map is generated by $E_{\ell -1} - 1$, any non-zero element of $\widetilde{M}_k$ must have filtration congruent to $k$ mod $\ell - 1$. (In short, the grading mod $\ell -1$ is well-defined.) So if $k < \ell -1$ then any non-zero element of $\widetilde{M}_k$ must have filtration equal to $k$.