i'm going to simplify, it, try to cut through the conflicting theories because i believe this is true:
Increasing strength over time is the most common way to get bigger, when it's within the context of moderate or higher reps that are increased over periods of time. That's not the only way, but it's the most common.
The main thing is progressive overload of the muscle, by increasing the intensity over time. Most commonly as i said above. Less common but still used is to keep the weight lifted relatively static and use other ways of increasing intensity over time again within the context of moderate or higher reps.
Increasing intensity can be increasing the weight, increasing the number of reps, the sets, reducing or increasing the rest between sets, adding intensity techniques like negatives, cheats, burns, forced reps, etc.
Typically you'll see a guy with big arms using relatively heavy weights on curls or preachers, or less commonly they've increased other types of intensity over time while using not so much weight, but even there guys with good development might have trained harder in earlie yeas and don't show the same intensity later. I have a feeling that's true of some of the top guys we see on training videos; they're coasting now relative to earlier days, thus someone can get the idea watching them lift now.
In my teens i used to watch a guy who placed in the top 3 of the Unverse doing preacher curls with 130, and that weight wasn't that difficult for him. Even guys who don't lift alot of weight are increasing intensity over time in some way-slight increases in weight, increasing reps, shortened rest periods, supersetting, going to negative failure with cheats, burns, negatives, etc.-it's all about increasing intensity.
Some look at powerlifters and think that weight increase doesn't equal size, but that doesn't apply when training in BB fashion within the context of moderate or high reps and shorter rests between sets. The powerlifter example confuses people, they think that means being stronger doesn't equate to size, but that's a different type of training.