In his book he says not to drink aspartame, coffee (caffeine), and not to eat nitrates (from what I understand is salt). Is he just giving this advice because they are unhealthy? They don't have carbs, so I don't see the relevence to a low-carb diet.

Coffee raises insulin and then drops it giving you sugar cravings. Some people still drink coffee (me included!) with no problems.
Aspartame is really bad for health (lots of research about it) and could give you sugar cravings, too. Some people are sensitive to all artificial sweeteners. Eating something "sweet" raises the insulin, thus giving cravings. Atkins suggests you use sucralose (Splenda) or sacharine.
Nitrates cause some disease, maybe cancer, but I'm not sure.
As always, each person is different, and you have to test what works for you and what doesn't.

Hope this helps and good luck!

Thank you!

Nitrates are A salt, but are not table salt. They are used for curing meat, and they have been implicated in some forms of cancer. Health food sources often have nitrite free deli meats.