Hi Corribus,
The washing soda is what shifts the grease, from wikipedia
Domestic use
In domestic use, it is used as a water softener in laundering. It competes with the magnesium and calcium ions in hard water and prevents them from bonding with the detergent being used. Sodium carbonate can be used to remove grease, oil and wine stains. It is sold as washing soda, soda crystals, or sal soda. Sodium carbonate is also used as a descaling agent in boilers such as those found in coffee pots and espresso machine.
I'm not a chemist, since high school, I just want to get the dishes clean without poisoning anyone. I take it that the result will not do that and it's safe to use as intended?