I heard on TV that you must remain VERY hydrated, i.e. drink
lots and lots of water when you are in Ketosis. They guy
gave no reason.

Can somoene please explain the detailed reason behind it?
Drinking plenty of water is always important. Particularly on a diet,
because people who don't drink enough water often end up eating more food so
that their body gets water from the food.

Does being-hydrated help you lose weight faster?
Yes

Does it give you energy so you won't give up?
Well, if you are dehydrated you won't have as much energy... so I suppose

Constipation is a common side effect of low-carb, and that can be helped by
making sure you are getting enough water.