To answer the question given in the title:
How can someone spoof texts with my phone number?
First, it is important to understand that one of the central systems phone providers use to communicate and to send information from one carrier to another (SS7) is based on trust. The system was developed in a time when all operators would trust the information received from others.
This gives way to sending information into a network with false content, something that is very easy, given that you have access to a SS7 gateway into the network. A carrier can simply tell another that the message is from a specific number, and this will be accepted.
More practically for the average Joe is to use internet services that give you the option to enter the desired sender's number, such as this. You are then able to send SMS messages pretending to be someone else.