Your package is on it's way to a USPS facility can also mean that your package has been accepted by the post office and then has been put on a truck and is now on it's way to the USPS facility such as a USPS sorting facility.
Some companies will have the USPS come by and pick up the packages and then when they do it should say accepted by USPS or something similar.
Then it should also say your package is on it's way to a USPS facility.
Either way when it says that your package is on it's way to a USPS facility it means that it is on it's way and could be that the package is on it's way to a local post office or another sorting facility.
Most commonly when the USPS tracking does say that your package is on it's way to a USPS facility it means that it's on it's way to a USPS sorting facility.
Then the package will go through the sorting facility and be loaded onto either a semi truck or airplane to go further to it's destination.
Yes sometimes the package may be shipped through another shipper such as FedEx, UPS etc and then they could update the tracking info to say it's on it's way to USPS facility.
Then USPS takes over the shipping and ships it to your address or local post office.