Given a compact Riemann surface $X$, every holomorphic function on $X$ is constant. This is obvious if you think about holomorphic functions as locally conformal mappings, that is, transformations of $X$ into a plane which locally preserve angles ("infinitesimal similarities"). Such a transformation is continuous, thus has a compact image, but the image is also open, so it must be constant: collapses $X$ into a point. In other words, the ring $\mathcal{O}(X)$ of holomorphic functions on $X$ reduces to the constants $\mathcal{O}(X) = \mathbb{C}$.
That's why, in the theory of compact Riemann surfaces, one is interested in meromorphic functions, i. e., those with at least one pole. These functions constitute a field denoted by $\mathcal{M}(X)$, and can be seen as branched coverings $X \rightarrow \mathbb{S}^2$ of the Riemann sphere/complex projective line.
One can show that there always exist non-constant meromorphic functions on $X$ (Riemann's existence theorem). This means that every compact Riemann surface is a branched covering of the sphere. This can be used to show that the field $\mathcal{M}(X)$ is a finite field extension of $\mathcal{M}(\mathbb{S}^2) =\mathbb{C}(z)$ (this last field is the field of rational functions = polynomial fractions).
Moreover, one can show that each finite extension of $\mathbb{C}(z)$ gives rise to a unique compact Riemann surface, up to isomorphisms, with the given extension as its field of meromorphic functions (Dedekind-Weber theory of algebraic function fields in one variable). This compact Riemann surface can always be realized as an algebraic curve in the complex projective space $\mathbf{P}^3(\mathbb{C})$ (without singularities, naturally). This means that it is the set of zeroes of some homogeneous polynomials with complex coefficients, in projective space.
One interesting question is to ask when a compact Riemann surface $X$ can be given by an equation with coefficients in $\overline{\mathbb{Q}}$ (the algebraic closure of rationals). It is known that this is the case if and only if there is a meromorphic function $X \rightarrow \mathbb{S}^2$ with at most three critical values (Belyi's theorem). A function of this kind is a covering of the sphere which is branched over three points (or less).
Thus, the study of compact Riemann surfaces/smooth plane algebraic curves over the algebraic numbers $\overline{\mathbb{Q}}$ is reduced to the study of coverings of the sphere branched over three points, which we can assume to be the points $0, 1, \infty$.
These branched coverings $f: X \rightarrow \mathbb{S}^2$ can be given a geometric representation. The fiber of $0$, is a finite set of points in $X$, which can be marked as black points. Points in the fiber of $1$ are usually colored white. The preimage of the interval $[0,1]$ (as a curve joining $0$ and $1$) is given by a set of curves joining black and white points, alternatively. This graph on $X$, formed by black points, white points and curves, is the dessin d'enfant associated to the branched covering $f: X \rightarrow \mathbb{S}^2$.
It is a remarkable fact that the branched covering $f: X\rightarrow \mathbb{S}^2$ determines the Riemann surface structure of $X$ (by pullback of the same structure in $\mathbb{S}^2$), and so, compact Riemann surfaces over algebraic numbers (with a specified meromorphic function branched over at most three points) are just orientable compact surfaces with certain graphs in them (the graphs associated with branched coverings).
Now, Grothendieck was interested in the Galois group of $\overline{\mathbb{Q}}$ over $\mathbb{Q}$, which acts on the coefficients of the equation of an algebraic curve over $\overline{\mathbb{Q}}$, giving another curve of the same type. Considering the dessins associated to those curves, that group then transforms a dessin into another dessin, and so, dessins can be used to obtain a geometric interpretation of the absolute Galois group.
