Dynare has an option options_.MaxNumberOfBytes governing how large a file on the disk is allowed to be. During the MCMC, Dynare will keep all draws in memory until that file size is reached and only then write the last draws to the disk. If you want to see what is going on with your chain before a full file is saved, you can use that option. I personally do not use this at all.
The _posterior_draws*.mat-files are used for the computation of moments_varendo. For the computation of these moments, you need the decision rules. Recomputing them each time for each new moment (e.g. variance, correlation, variance decomposition, etc.) is impractical. So we save both the draw (first column) and the computed decision rule (second column) in the cell array for each parameter draw (rows).