From the viewpoint of classical algebraic geometry, the reason is simple: they are easy to deal with and many things can be computed on them (e.g., their moduli, Picard lattices, automorphism groups, etc). For example, complete linear systems on projective K3 surfaces are particularly easy to study using results of Saint-Donat and Nikulin. Moreover, many special K3s turn up naturally in other problems from algebraic geometry e.g., as complete intersections (and this is not so much the case for other 'exotic' surfaces like Enriques surfaces). This makes K3 surfaces a nice testing ground for results in algebraic geometry.

Some examples:

-- Let $D$ be an effective divisor on a K3 surface. Then $|D|$ has no base-points outside its fixed components.

-- K3 surfaces have nice vanishing theorems: The vanishing of $h^2(X,D)=\dim H^2(X,O(D))$ for $D\neq 0$ effective is immediate by Serre duality: $h^2(X,D)=h^0(X,-D)=0$.

-- Let now $D$ be an effective nef divisor (so that $D.C\ge 0$ for every curve $C$). If $D^2>0$, then either $|D|$ is base-point free, then $h^1(X,D)=0$ and the generic member of $|D|$ is smooth and irreducible, or $|D|$ is not base-point free and then we have $D\sim kE+\Gamma$, where $k\geq2$, $E$ is an elliptic curve, $\Gamma$ is a rational curve and $E.\Gamma=1$. If $D^2=0$, then $| D|$ is composed with a pencil, i.e $D=kE$, where $E$ is an elliptic curve and $h^1(X,D)=k-1$. In sum, if you have a divisor that is free from fixed components, then you can calculate the dimensions of all the cohomology groups using Riemann-Roch.

-- Moreover, Saint-Donat gives precise results on the degrees of the generators and relations of the section ring $A=\bigoplus_{n\ge 0}H^0(X,nD)$. This means that one can easily study concrete projective models of K3 surfaces.

-- There are also strong results by Kovács on the effective cone of a K3 surface.

hardenough to deal with compared to curves, rational varieties and abelian varieties to have stood as a challenge to leading algebraic geometers for the last 60 or more years. (Weil's 1958 coining of the term K3 surface references the K2 peak -- i.e., very hard to master.) But I would also say that the love affair with K3 surfaces is neither especially recent nor especially particular to K3 surfaces: algebraic geometers have been interested in lots of algebraic varieties for some time now... – Pete L. Clark Mar 23 '11 at 21:49sheavesare supposed to be related to or analogous to the moduli spaces ofcurvesthat are considered in, e.g., Gromov-Witten theory... (Very rough idea: consider ideal sheaves of dimension 1 subvarieties of your variety, rather than maps of curves into your variety.) – Kevin H. Lin Mar 24 '11 at 2:07