Budgeting is used to help you spend the right amount of money you want. So, lets say you plan to buy a house your budget is 100,000. 100,000 is the amount of money you want to spend on buying that house. Budgeting plays a major role in planning it helps you not spend and not go over board on buying something. To sum up, the role budgeting plays in planning is to "make sure you don't spend over the amount of money you want to spend on something, it's your limit when dealing with the amount of money you want to spend."
Hope this helps!