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.
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 $$0\to O_X\to \mathcal{L}\otimes L^{-1}\to O_X\to 0\in Ext^1(O_X,O_X)$$