How does the checkbox to 'Apply to all Store Integrations' work on the price override dialog?

If you check this box for any SKU, this will automatically update the price override in each integration for that SKU. 

For example, if you have the following two integrations:

Supplier A > Shopify
Supplier A > BigCommerce

Then if you are overriding the price under Shopify integration for that SKU and you also want to update that SKU in Bigcommerce integration as well, by checking this box, you do not need to manually go and change it in the other integration. 

Checking this box will automatically update the SKU price in both integrations.