You need to distinguish between "coproduct" and "comultiplication." The categorical coproduct is just a generalization of addition and is intuitive in many contexts.
Comultiplications are more interesting. Actually it turns out that many familiar mathematical objects are canonically equipped with comultiplications. In fact, in any category $C$ with finite products every object $c$ has a canonical comultiplication given by the diagonal map $\Delta : c \to c \times c$. On sets this is the diagonal map $x \mapsto (x, x)$. In other words, in such a category $C$ every object is canonically a coalgebra. Usually this structure is invisible, but coalgebras are preserved by monoidal functors, so for example the free vector space on a set is canonically a coalgebra in vector spaces. Usually this structure is also invisible, but it appears for example in the Hopf algebra structure on group rings...
In particular, every topological space is canonically a coalgebra. Now, homology (over a field) is a covariant monoidal functor, so it sends coalgebras to coalgebras. But cohomology is a contravariant monoidal functor, so...
Another source of comultiplications is combinatorics. Where multiplications are intuitively like "building things up," comultiplications are intuitively like "breaking things down." And this allows you to define many natural coalgebras by breaking down combinatorial structures into parts. For example, $R[x]$ has a natural $R$-linear comultiplication given by
$$x^n \mapsto \sum_k x^k \otimes x^{n-k}$$
and this reflects the process of breaking apart a string of length $n$ into two strings of length $k$ and $n-k$. The dual of this coalgebra is the algebra of ordinary generating functions over $R$. Similarly, there is another comultiplication given by
$$x^n \mapsto \sum_k {n \choose k} x^k \otimes x^{n-k}$$
and this reflects the process of breaking apart a set of size $n$ into two subsets of size $k$ and $n-k$. The dual of this coalgebra is the algebra of exponential generating functions over $R$.
In the same spirit, Rota noticed that the incidence algebra of a poset is naturally regarded as the dual of an incidence coalgebra, namely the free vector space on intervals $[a, b]$ in the poset with comultiplication given by
$$[a, b] \mapsto \sum_{a \le c \le b} [a, c] \otimes [c, b]$$
and this reflects the process of breaking apart an interval into two subintervals.
Yet another source of comultiplications comes from objects in a category $C$ which represent a (covariant) functor to the category of monoids. For example, the fundamental group functor $\text{hTop}_{\ast} \to \text{Grp}$ is represented by the object $S^1$ in the (pointed) homotopy category, and this is because $S^1$ naturally has a cogroup structure. Similarly, the multiplicative group functor $\text{CRing} \to \text{Grp}$ is represented by the object $\mathbb{Z}[x, x^{-1}]$, which also naturally has a cogroup structure (in fact a commutative Hopf algebra is precisely a cogroup object in $\text{CRing}$, or equivalently is precisely the ring of functions on an affine group scheme).
Comultiplications only seem unfamiliar because they were historically not pointed out, but actually they are all over the place. See also What is a coalgebra intuitively?
