2 added 204 characters in body

From the viewpoint of classical algebraic geometry, the answer reason is simple: they are easy to deal with and a lot of 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-Donant 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 birational 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 $| D|$ is base-point free, $h^1(X,D)=0$ and the generic member of $|D|$ is smooth and irreducible. 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-Donant 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.

-- 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 $| D|$ is base-point free, $h^1(X,D)=0$ and the generic member of $|D|$ is smooth and irreducible. 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-Donant 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.