As shown in the Wikipedia page on the flatness problem, you can easily show from the Friedmann equations that the density parameter $\Omega$ is related to the energy density $\rho$ and scale factor $a$ by
$$\left( \Omega^{-1} -1\right)\rho a^2 = -\frac{3 \kappa c^2}{8\pi G}\ ,$$
where $\kappa$ is the curvature parameter, and is zero, $+1$ or $-1$ for a flat, closed or open universal geometry.
$\Omega$ is the ratio of $\rho$ to the critical energy density - the density that would just halt the expansion of a flat universe with no dark energy. I believe the plot you show imagines such a universe and $\Omega$ determines the time evolution of the scale factor - if $\Omega >1$ the universe ends in a big crunch, if $\Omega <1$ it expands forever.
The RHS of the equation above is constant (zero in a flat universe) and therefore the LHS also must be a constant. Now in principle $\Omega$ varies with the scale factor and because the density $\rho$ is proportional to $a^{-4}$ (early, radiation-dominated universe) or $a^{-3}$ (later, matter-dominated universe) then we can either write
$$\left(\Omega^{-1}-1\right) \propto a^2\ \ \ {\rm radiation}$$
$$\left(\Omega^{-1}-1\right) \propto a\ \ \ {\rm matter}$$
In either case we see that if $\Omega$ differs from 1 then that difference is amplified as $a$ becomes larger.
That is what you are seeing in the plot. An early universe with $\Omega>1$ by a tiny amount, grows into a universe with $\Omega \gg 1$ as it expands. Similarly, an early universe with $\Omega < 1$ by a tiny amount grows into a universe with $\Omega \ll 1$. Only an early universe where $\Omega$ was very, very close to 1 can grow into the universe with $\Omega \sim 1$ that we see today.