What do web developers mean when they think something will make their site look "busy"?
What gives this feel of busy-ness?
busy - biz -ee:
Too many elements, or badly distributed ones,(it's possible to design with many elements and yet don't make it look busy, while area can look busy with just 3 elements) might destroy a design, its functionality, distract too much and make it confusing. And also what you wanted to be the main focus might loose strength. Also, busy designs give an unpleasant feeling, which is far from the purpose(in most cases). This can be applied to many other kind of designs, not just web sites. (And... can be really terrible that obsession of killing all sort of empty space, asked to remove it even when being used with a design purpose, as if it were bad no matter how you'd use it... How was the latin term... "horror vacui")
I think the other answers correctly identify the symptoms, but for me the underlying cause is a lack of structure in the information presented, and the best way of resolving that it is to define some kind of hierarchy.
You can start with questions like:
From there you can start to look at visually differentiating elements or classes, using all the tools at your disposal - grids, colour, white space etc - but you have to do the classification first.