With 3 indicators one factor, DF=0, SRMR=0. As the number of indicators increases, DF increases, the factor model has to fit more and more correlations (quadratic growth for the number of parameters that have to be fitted) with linearly increasing number of parameters. So one would expect bigger differences as DF increases. BTW, this is not the MSE for the correlations, rather sample v.s. estimated values both of which vary across samples.
The second reason is that the zero residual parameters (means and variances) have a higher weight with smaller number of indicators.
The SRMR dependence on DF is relatively small though.