If $X$ is a metrizable space, the topology of $X$ is completely determined by the convergent sequences: if we know which sequences in $X$ are convergent, and what their limits are, we can determine exactly which subsets of $X$ are open. This is actually true in a somewhat larger class of spaces than just metrizable spaces, but it is not true for topological spaces in general.
For example, let $X$ be an uncountable set, let $\tau_0$ be the discrete topology on $X$, and let $\tau_1$ be the co-countable topology on $X$. Then $\langle X,\tau_0\rangle$ and $\langle X,\tau_1\rangle$ are not homeomorphic, but they have exactly the same convergent sequences. If $Y$ is any uncountable subset of $X$ such that $X\setminus Y$ is also uncountable, then $Y$ is open in the discrete topology but not in the co-countable topology. And in each topology the convergent sequences are precisely the sequences that are eventually constant.
Nets are a generalization of sequences powerful enough to capture the topology of any space, not just metrizable spaces: if $\tau_0$ and $\tau_1$ are topologies on a set $X$ that have exactly the same convergent nets, then $\tau_0=\tau_1$. This, very simply, is the main motivation for looking at them.