Suppose $X/\mathbb{Q}$ is a (smooth, projective, geometrically integral) curve of genus $g\geq 2$ and $J/\mathbb{Q}$ its Jacobian variety.
If one is interested in determining the (finite, by Faltings) set of rational points $X(\mathbb{Q})$, then it can be useful to compute $J(\mathbb{Q})$ first. The latter is easier because $J(\mathbb{Q})$ is a finitely generated abelian group, and descent theory analogous to elliptic curves allows us to often do this in practice.
If we pick a point $P\in X(\mathbb{Q})$ then we have an associated embedding $i_P: X \hookrightarrow J$. In favorable situations studying this embedding allows us to determine $X(\mathbb{Q})$ from $J(\mathbb{Q})$.
For example, the method of Chabauty-Coleman gives a very concrete instance of this when the rank of $J(\mathbb{Q})$ is less than $g$ (for a friendly introduction to this method see the nice survey of McCallum-Poonen).

The moral is: by replacing $X$ by $J$, we somehow have made the geometry harder but the arithmetic easier.

The relation with motives can be explained in relatively concrete terms. The $\ell$-adic cohomology groups $H^i(X_{\bar{\mathbb{Q}}},\mathbb{Q}_l)$ are zero if $i\neq 0,1$, $2$ and isomorphic to $\mathbb{Q}_l, \mathbb{Q}_l(-1)$ if $i=0, 2$ respectively. (The minus $-1$ denotes the Tate twist.)
So the only interesting degree is $i=1$, and pulling back via $i_P$ will induce an isomorphism $H^1(X_{\bar{\mathbb{Q}}},\mathbb{Q}_l) \simeq H^1(J_{\bar{\mathbb{Q}}},\mathbb{Q}_l)$. This last group (with its Galois action) is isomorphic to the dual of the $\ell$-adic Tate module of $J$.
So $J$ and its torsion points encapsulate all the cohomological information of $X$.
Similar statements will hold for other Weil cohomology theories: the only interesting degree is $1$ and $i_P$ will induce an isomorphism on $H^1$.

Edit: as pointed out in the comments, the geometry of $J$ is arguably easier than that of $X$. A better moral is thus maybe that we have made the space we're considering larger but richer in structure.