The inspection date is the first date when they recommend doing a visual inspection and taste test to make sure the MREs are still good. This date is usually three years after the MREs are produced (date packed).
The reason they do this is because even though MREs are supposed to last a long time, their quality can vary based on how long they've been stored and under what conditions (temperature, mainly) they were stored.
So they do this inspection and if the MREs still look good, they keep them around for a while longer. If not, they destroy them.