Here is my impression ...
(I am very much a non-expert in the physics (and probably the mathematics too) so I may well be wrong about some of these things.)
Algebraic geometry sometimes enters the picture in string theory and physics because, while we start with, say, a compact Kähler manifold, for some reason or another we maybe get an integral Kähler class (for example see this MO question), and thus our manifold is projective by the Kodaira embedding theorem, and thus it is algebraic by Chow's theorem. Conversely, we may be actually interested in possibly non-algebraic compact Kähler manifolds in the physics or string theory, but the algebraic manifolds will provide at least a pretty big class of nice examples to play with.
And at least for smooth projective algebraic varieties, GAGA theorems tell us that many things (like for example, sheaf cohomology) are the same whether we consider our space as an algebraic variety or as an analytic thing. For the B-model side of mirror symmetry, I think this is how algebraic geometry (as opposed to complex analytic geometry) generally comes into play --- via GAGA theorems or at least "GAGA principles". For example, it is a fact that analytic coherent sheaves on smooth projective varieties are algebraic. From this it follows that, at least for smooth projective varieties, the derived category of coherent sheaves is the same whether we look at things algebraically or analytically. (I'm guessing, but I don't know for a fact, that in the physics the analytic objects are the a priori relevant ones.)
Another interesting issue is the fact that algebraic geometry often appears even on the A-model side of mirror symmetry, which is supposed to be the symplectic side of the story. I don't really know anything about this, so maybe someone else can say more, but there's some work on, for example, the relation between the symplectic version of Gromov-Witten theory and the algebraic geometry version of Gromov-Witten theory -- they're supposed to coincide in the case of smooth projective varieties. It's perhaps not too surprising, since the symplectic version of GW theory involves J-holomorphic curves after all, but it's definitely not a trivial result.
I suppose the naive explanation for the appearance of surfaces is that they're worldsheets of strings, but I don't really know the explanation for why the surfaces should have complex structures, i.e. why they should be Riemann surfaces (and by the way, it is also a basic fact that any compact Riemann surface is actually algebraic) nor do I know why the maps from the curves to the target manifolds should be holomorphic or J-holomorphic. I hope that other MO users, especially people who know about string theory and physics, can say more about these things...
