The weight limit is the maximum that a fighter can weigh at the weigh-in before the fight. There is nothing within the rules to prevent a welterweight from fighting a light heavyweight, as long as he is below the weight limit of 175 lbs. There are, of course, practical reasons why this sort of thing doesn't happen often. The weight classes exist for a very good reason, namely that a larger man has a distinct advantage over a smaller man in this sort of sport. The weight classes are an attempt to control for this.
The size advantage can be considerable. Over the years, the weigh-in has moved further and further from fight time in an attempt to prevent death and injury in the ring due to dehydration. This dehydration came about because larger fighters would do such things as stop drinking water 48 hours before the weigh-in, take diuretics, or work heavily while dressed in a plastic suit in order to lower their body weight. Using these techniques, a 160 lb fighter, (a natural middleweight) might be able to lower his weight to 145, and thus qualify to fight as a welterweight. The major problem with this is that such drastic fluid loss makes a fighter more vulnerable to cerebral edema, and greatly increases his risk of death in the ring. In an attempt to prevent this, the weigh-in has been moved further away from actual fight time so that fighters that do this have a chance to rehydrate. And as a result of this, fighters who weigh in at 146 1/2 might actually enter the ring at 155, or 158, or 160 lbs.