When someone submits an expense to your Project, the Core Contributors will get an email notification. The email will show all the details of the expense, like who submitted it, the amount, and the receipt or invoice.
You can also view expenses on Donate PR. Click the link from the email notification, or go to your Project page and click "View all Expenses".
The decision is up to you as a project. Some projects are informal, and any Core Contributor can approve any expense. Others have a formal decision-making or budgeting process, or only pay expenses for specific things. We advise you have a discussion with your collaborators and agree guidelines for approving expenses.
When you approve an expense, the submitter will get a notification. The Fiscal Host admin will also be notified. They review the expense and make sure it has a valid receipt or invoice, and then proceed to pay it.