Perhaps this might work: you understand that in an abelian category the coproduct and the product of two objects are the same (when they both exist). You want to "understand" something about the coproduct and the product of infinite families and how they may differ.
In an abelian category, if $J\subseteq K$ are finite, then you have maps from $\prod_{j\in J}X_j$ to $\prod_{k\in K}X_k$ and back; the map from the $J$-indexed product to the $K$-indexed product is obtained by "extending by $0$s", while the map back is obtained by dropping the components not in $J$. These are in fact maps induced by the universal properties of the product and the coproduct by the canonical projections and immersions, recalling that the finite products are equal to the finite coproducts.
That means that given an infinite family $\{X_i\}_{i\in I}$, you can look at the family of finite products $\bigl\{\prod_{j\in J}X_j\mid J\subseteq I,\ |J|\lt\infty\bigr\}$ as both a directed family (ordered by the partial order of inclusion among finite subsets) and as an inversely directed family. If you take the direct limit of this family and interpret the objects as coproducts, then you get the coproduct of the full infinite family. If you take the inverse limit of this family and interpret the objects as products, you get the product of the full infinite family.
The direct limit of the objects is defined by "local" conditions, so that intuitively you only need to worry about what is happening "locally", hence the objects should only require finitely many information; whereas the inverse limit requires a whole family of consistent objects, so that you need "a lot more" information. Intuitively, local information just isn't enough.
For abelian groups, modules, and vector spaces, the directed limit consists of equivalence classes of objects of the form $[(x,J)]$ with $x\in \prod_{j\in J}X_j$, with $[(x,J)]=[(y,K)]$ if and only if there exists $M$ finite, $J,K\subseteq M$, and such that the images of $x$ and $y$ in $\prod_{m\in M}X_m$ agree; so elements are completely determined by information on a finite subset of $I$.
On the other hand, the inverse limit is constructed at the set level as a subset of the set-theoretic product $\times(\prod X_i)$ of the families, taking those elements $(x_J)$ indexed by the finite subsets of $I$, such that if $J\subseteq K$ then the image of $x_K$ in $\prod_{j\in J}X_j$ is precisely $x_J$ ("consistent families"). This means that you may need information about what is happening on all coordinates to determine an element, and there are elements that are not obtained merely by local information.
(A possible model might be the difference between the directed limit of the cyclic groups $\mathbb{Z}_{p^n}$ and the inverse limit of the same groups; the directed limit is the Prüfer group, in which each element is, in a sense, "finite", whereas the inverse limit is the $p$-adic integers, in which you have elements that are not "finite", but rather "come" from 'infinite tuples'.)