Here's a breakdown of the flow for manual service approval:
Under [Packages] -> [Order Forms], every order form should have the option checked for "Require Manual Review and Approval of All Orders". When a user places an order, whether they pay for the service or not, the order is "In Review", and no action is taken to provision the services ordered. To manually approve an order, you mark it as "Accepted" in the Order widget under [billing] -> [Overview]. The status then changes from "In Review" to "Pending". When the invoice for the order has been paid, if not already, it will be activated. The status will then change from "Pending" to "Active".
It's important to note the subtle difference between "Pending" and "In Review" statuses. A pending service can be activated automatically once paid while one in review cannot.
If this is not how orders are working for you, then there is something amiss, and we'd need to know more about your settings, and version of Blesta, to help diagnose the issue.