It’s actually pretty easy, if you understand whet grout is. It’s essentially fat from your skin, that picks up grime – hence the unsightly black colour.
As with any natural oil or fat, the answer is caustic soda.
So you have two choices:
1. Use an oven cleaner
2. Make your own caustic soda solution. Get a generic 1L pump-spray pack from Bunnings (the type that you pressurise by pumping the knob on top up and down. Put half a cup of caustic soda flakes in a cup, then empty into a kitchen jug. Add water until the caustic soda dissolves (it might get a bit hot) and then add it to the spray pack and make it up to volume with water. Hey presto, you have a instant oven cleaner.
Now all you do is spray a generous amount onto the dirty grout and leave it for half an hour. Then come back to it with a stiff brush and you’ll find the grout will come away very easily. With just the slightest pressure it will come away and there’ll be none of the vigorous scrubbing that you’re used to