I won't claim this means ''understanding mirror symmetry'', but if you are familiar with the derived category of coherent sheaves, then there is a consequence of Kontsevich's homological mirror symmetry that can be understood, without knowing anything about the Fukaya category:
For every symplectic diffeomorphism of the mirror $\hat X$, there is a autoequivalence of $\mathrm D^b(X)$.
Examples:
If $X$ is a an elliptic curve, then $\mathrm{SL}_2(\mathbb Z)$ acts as a group of symplectic diffeomorphisms on the mirror $\hat X = \mathbb{R}^2/\mathbb{Z}^2$. There is a corresponding action of (a central extension of) $\mathrm{SL}_2(\mathbb Z)$ generated by Fourier-Mukai transform induced by the Poincare line bundle on $X \times X$, and by tensoring by a line bundle of degree one (and shifts).
A Dehn twist corresponds to the ''spherical twist'' $\mathrm{ST}_E$ at an spherical object $E$ (see arXiv:math.AG/0001043).
More examples have been studied by Horja, see arXiv:0103.5231.
