I've been lifting consistently for over a month and have noticed that i'm gaining weight (unfortunately) but you wouldn't know by the way my clothes are fitting. I would think that the extra weight would be causing my clothes to be tighter. I've heard muscle weights more than fat but the doesn't seem logical...a pound of muscle and a pound of fat still weigh a pound. It seems a contradiction to gain weight and clothes to fit better, what is the reason behind this?
You're right that a pound of fat and a pound of muscle both weigh a pound, but a pound of muscle takes up less space than a pound of fat. So, if you have the same volume of muscle and fat, the muscle will weigh more. If you're working out a lot, you may be toning up and getting smaller but still gaining weight - this is because you're converting fat to muscle, which takes up less space. This article puts it pretty clearly: "muscle weighs more by volume than fat."
First converting fat into muscle is a myth, it's physiologically impossible. Fat is made out of Adipocyte while muscle is made out of mostly proteins.
You can improve your body composition by increasing your muscle mass while decreasing your body fat but the two are happening independent of each other.
Another myth is toning. Again very similar to muscle -> fat. It's possible to achieve a 'toned' look but I prefer not to use that terms since its misleading.
As for the reason why you are gaining weight from strength training. It comes down to several factors:
Inflammation and water retention: lifting causes trauma to the muscle and skeletal systems which causes an inflammatory response in order to repair the damage. This leads to hypertrophy and anabolism if you eat and rest sufficiently. More specifically this causes water retention which increases your bodyweight.
Increased glycogen storage: any type of anaerobic lifting uses the immediate energy supply found in the muscle which is glycogen. Lifting encourages the body to store more glycogen so it's better prepared for future activity. This again increases your overall bodyweight.
- Increased lean mass: lifting will cause you to gain more lean body mass. This isn't simply increased muscle mass but also increase in the systems that support the muscle like connective tissue. For someone on a caloric deficit you would be lucky to gain .25lbs per week, most likely less.
So there you have it, those are all the factors which impact your weight in relation to training.