Create the TimeTracker users in your org by following the instructions here. And add the Expense Manager to the user in the Expense Manager field.

Once the Expense Manager is added to the user, the Expense manager sees the expenses requested by the user and can approve or Reject the user expenses.

Note: Even the Expense Manager should be created as TimeTracker user in Salesforce.  Only then you can add them as Expense Manager for other users and they can Approve or Reject the user Expenses. Click here to know more about creating a Expense Manager ID.