It does not mean low carb per se, it means: low sugar combined with low fat!
That right there is absolutely key. Complex carbohydrates and small amounts of sugar consumed with lots of fiber are perfectly healthy. Starchy low-fiber foods are almost as bad as sugar (they're broken down into sugar very rapidly by the digestive system) for you, and sugar is outright terrible if it isn't properly moderated.
If you eat carbs that can easy be converted into sugar *plus* a lot of fat, the fat is not digested but stored in your fat cells.
It's worse than that. If your insulin levels spike high enough your fat cells will happily draw in sugar and make it into fat for long-term storage. Eat all sugar all the time and you'll get just as fat as if you eat a mix of fats and sugars.
You're actually better off eating more fats than carbs (all things in balance, of course, carbohydrate-rich foods are great sources of the nutrients we need) since our bodies have evolved better regulatory systems to deal with fats than starches. And if you're going to eat a lot of carbs, make sure they're in a form that comes with a good amount of fiber, since it slows down digestion and reduces the sock of sugar your system would otherwise receive.
That is a simple matter of insulin level in your blood.
Insulin is probably the most important hormone when it comes to regulating hunger (I've heard this extended to mood in general, though I think that's reaching), metabolism, fat production and storage, and some stages of digestion. Fuck up your insulin levels bad enough for long enough and you'll become fat, lethargic, and eventually diabetic.
Avoid foods which are high in sugar (or starch) and low in fiber. That combination is very common in modern cuisine, but it's been exceedingly rare for the bulk of our species' history, and we're not equipped to gracefully handle it. The digestive system processes the carbs too quickly, and that makes the pancreas overreact with insulin production. The hour or so following the sugar rush gets spent in a state of mild starvation since blood sugar has dropped faster than the body can remove the absurd amount of insulin it had to make (which means that the fat cells spend that time hoarding energy instead of releasing it for consumption) to deal with the sugar spike. The brain notices the low level of energy available in the bloodstream and response by switching hunger back on, despite the fact that the body's energy reserves would soon be released for consumption as insulin levels fall off.
Desserts are awesome, of course, and not much is going to go wrong with you if you indulge in moderation. But if you do this constantly (constantly eating lots white breads, pastas, cookies, drinking sweetened drinks, etc), then you'd better be getting a ridiculous amount of exercise to offset the damage you're doing yourself.
(Of course, people get fats for lots of other reasons besides sugar consumption. I'm not your doctor, I'm just some guy who's had to take care of diabetic friends and family members. If this advice doesn't work for you, see an actual doctor or dietitian. Imagine some other standard disclaimers here.)