Does having much knowledge in other branches help one's research in mathematics in one's own field?
The answer is yes. It helps. Mathematics is very broad. In the various major branches, such as Analysis, Algebra, etc. different insights are needed to really understand what is going on and what is important. In addition to the all-important insights, different proof techniques are common. 

When you are doing real research in mathematics you are engaged in a very narrow study. But having a wide variety of ways to attack a problem, even if they might be suggested by ideas from another subfield, will give you an advantage. 

Both breadth and depth are needed. Depth is usually what is most valued, but the breadth of knowledge helps you get there. So, don't completely neglect other mathematical subfields than the one you are most interested in. You don't need to be equally skilled in all of them, but being able to grasp the essential insights is a plus.