The assignment of budgets can be done from the "Iterations" and "Times Explorer" pages.
From the "Iterations" page, you can assign a budget to iterations or items. You can can bulk assign budgets to one or many time records to override the inherited budget or set budgets to items without a budget. On the "Times Explorer" page, you can then check on every single time record and what budget it will be counted in.
Adding Objects to a Budget
The "Budget" page allows you to directly add objects, like iterations and work items, to the "Included" and "Excluded" list of items.
Included list of Items
The items in the "Included" list indicates that for all these items, the tracked time is calculated in that budget. So, when you select the budget, the Tracked field will display the total time tracked for all included items.
Excluded list of Items
The items in the "Excluded" list indicates that for all these items the tracked time is not calculated in that budget. So, when you select the budget, the Tracked field will not include the tracked time for all excluded items. Usually, you can use the excluded items list to remove certain items from the budget. For example you can include a parent iteration or a PBI in the budget but exclude certain child iterations or child work items from the budget.