Let Nil be the subgroup of $GL_3(\mathbb{R})$ given by matrices of the form $ \left( \begin{array} 11 & x & z\\ 0 & 1 & y\\ 0 & 0 & 1 \end{array} \right) $ with $x,y,z \in \mathbb{R}$. The collection of matrices with $x = 0$ comprises a normal subgroup isomorphic to $\mathbb{R}^2$, and so there is a short exact sequence $ 1 \longrightarrow \mathbb{R}^2 \longrightarrow \text{Nil} \longrightarrow \mathbb{R} \longrightarrow 1; $ in other words, Nil is an extension of $\mathbb{R}$ by $\mathbb{R}^2$.
I have been told that this fact explains why Nil does not contain a rank-two free group as a subgroup, but I have not been able to piece together the details of the argument. Can somebody help me out here? Generally, if $G$ is an extension of $H$ by $K$, under what hypotheses on $H$ and $K$ can one conclude that $G$ does not contain a rank-two free group?