In general, unlike say singular homology, constructions in homological algebra are usually not all that geometric. Nevertheless they do sometimes translate into geometry. Let's consider the simplest case, where both subvarieties are $X$ itself. Then you are asking what does $Ext^1(O_X,O_X)\cong H^1(X,O_X)$ mean geometrically? One answer is that it is the tangent space to the Picard variety (or more correctly scheme) at some given line bundle $L$. In fact, the $Ext$ interpretation gives an nice way to see this. Let $k$ denote the ground field. Then a tangent vector to $Pic(X)$ at $L$ is just a first order deformation of $L$, i.e. a line bundle $\mathcal{L}$ on $\mathcal{X}= X\times Spec\ k[\epsilon]/(\epsilon^2)$ which restricts to $L$ on $X$ viewed as subscheme of $\mathcal{X}$. It follows that there is an exact sequence $$0\to \epsilon O_{\mathcal{X}}\otimes\mathcal{L}\to \mathcal{L}\to L\to 0$$ which can be identified with $$0\to L\to \mathcal{L}\to L\to 0$$ This in turns gives an extension $$0\to O_X\to \mathcal{L}\otimes L^{-1}\to O_X\to 0\in Ext^1(O_X,O_X)$$
Perhaps I can do one more case, which may be more typical. Say $Y$ and $Z$ are curves on a smooth surface $X$ with no common components. Then relevant $Ext$ group is easy compute using some standard tools from homological algebra. The so called local to global spectral sequence implies that $$Ext^1(O_Y, O_Z) \cong \bigoplus_{p\in Y\cap Z} Ext^1_{O_{X,p}}(O_{Y,p}, O_{Z,p})$$ The latter is just the sum $$\bigoplus_{p\in Y\cap Z}O_{X,p}/(f_p,g_p)$$ where $f_p$ and $g_p$ are the local equations for these curves. I admit however that I haven't thought about what this means in terms of extensions.
