When processing orders from BigCommerce to Shiptheory, depending on which trigger you've set Shiptheory to use in BigCommerce, will depend on when the order/shipment will be appear and be processed by Shiptheory.
You can change the status in the BigCommerce settings in Shiptheory, as seen below:
For an order to come in, just change the status in BigCommerce of the order(s) you wish to process.
The order will be sent to Shiptheory and if it matches a rule, it will be booked in with the corresponding carrier/service. The tracking number will be sent back to BigCommerce and the label will be available to print, or print automatically, depending on your printing setup.
You'll be able to see the Shipment in Shiptheory if you need as well as in BigCommerce.