There is a very nice paper exactly on this topic, where the expressions describing the fit curves are derived: http://arxiv.org/abs/hep-ph/9906447v1 If you have a look at the final expression (formula (6.5) on page 8), you will agree, that the relation between $\Omega_\Lambda$ and the luminosity is hard to describe by words.
However, you can try to think about this in the following way: Comparing an accelerated universe with a non-accelerated universe which at present time have both the same size (or, to be more precise, the same value of the scale factor $a_0$) and the same expansion ratio (or, to be more precise, the same value of the Hubble constant $H_0$), we can ask ourselves, which one was smaller at a given time in the past, say 10 billions of years ago. If you think about it, you will come to the conclusion, that an accelerated universe was larger 10 billions of years ago than a non-accelerated universe. You can have a look at the graphics from figure 2 on page 5 of the paper linked above and compare graphics for different values of $\Omega_\Lambda$. For instance compare the $y$-values of the universes $A$ and $E$ at early times. $A$ has $\Omega_\Lambda=0$ while $E$ has $\Omega_\Lambda=0.9$. As you can see the size of $A$ is smaller than the size of $E$ as the curve of $A$ is below the curve of $E$ for all $t-t_0<0$.
If you agree to this, the rest is easy. Given a SNIA that emitted a photon some billions of years ago, we know that the ratio $a_0/a(t)$ of the universe's size at the time of emission and today is smaller in an accelerated universe than in a non-accelerated one. Therefore, the redshift of the light will also be smaller, since the redshift $z$ is given by $1+z=a_0/a(t)$. Therefore, for given apparent magnitude (i.e. also given the current distance), the points in the graph that you linked in your question will be shifted to the left in an accelerated universe as compared to a non-accelerated one.