Let us consider a blob of fluid in a thin Keplerian disc. If the fluid element is "squished" in the radial direction, it might collapse to a clump, or return to its unperturbed state. The time scale for returning to its original shape is the inverse of the epicyclic frequency. For simplicity, we will use the Keplerian angular velocity $ \Omega $ instead. The time scale for collapse is just the Jeans' time $ \frac{1}{\sqrt{G \rho}} $, where $ G $ is the universal constant of gravitation, and $ \rho $ is the density. The criterion for stability is therefore that the period is much shorter than the Jeans time

$ \frac{1}{\Omega} < \frac{1}{\sqrt{G \rho}} $

The volume density is just the surface density divided by the height $ \rho = \frac{\Sigma}{h} $. The height is determined by pressure balance with gravity

$ \frac{h}{R} = \frac{c_s}{v_k} \Rightarrow h = \frac{c_s}{\Omega} $

where $ R $ is the distance from the centre of the disc, $ c_s $ is the local speed of sound, and $ v_k $ is the Keplerian velocity. Putting it all together yields that a blob is stable if

$ \frac{c_s \Omega}{G \Sigma} > 1 $

The region of the disc most susceptible to the Toomre instability is its outer edge. At that point the surface density is roughly given by $ \Sigma \approx m_d / R^2 $, where $ m_d $ is the mass of the disc. The Keplerian velocity is $ v_k \approx \sqrt{G M/R} $ where $ M $ is the mass of the star around which the disc rotates. Substituting these relations we obtain a simplified version of the Toomre stability criterion

$ m_d < \frac{h}{R} M $