In the Bigcommerce app marketplace locate the Shiptheory app:

Click the Shiptheory app and click Install:

This will then show you a confirmation page.  Once you are ready, click Confirm:

If you are currently not logged into Shiptheory or don't have an account you can either sign in or create an account and sign in:

You'll then be directed to the activation settings.  Here you can specify when you process orders, e.g. when a status update will trigger the shipments being sent over to Shiptheory:

Once selected, click Activate:

You will then be directed to your settings where you can specify on which status you want Shiptheory to process orders:

When you're done, click Save.

Other relevant guides

Processing Orders and Shipments from BigCommerce

Shipping Rules Explained

Auto Printing

Configuring BigCommerce & Shiptheory

Specifying the number of Boxes in BigCommerce & Shiptheory