Budget groups allow you to organize budgets - for example, you can group household-related budgets together, kid-related in a separate group, etc.
To create a budget group, you should click on the budget group icon at the bottom of the budgets list (the middle one).
You'll be asked to indicate the group name and group color. You'll also be able to select which accounts to put in a group by selecting them from the list.
Please note that it's not possible to group expense and income budgets together.