Just add a new table to the database to store the key. You may want to create a new column if you don't want to add a new table. In that column, you should store the key that is going to be sent for the verification purposes.
Then send the key to their phone numbers. You would need to use a gateway service to send the messages through SMS to their mobile phone. You can use services like Twilio or so on. PHP would just be used to make things work.
Now if you have done what I told you previously (in previous 2 paragraphs) then you are just going to ask them for their phone numbers and then send them a verification code. Verification is just like authentication. You are going to test their phone number and verification code, if the code is for the current phone number, then you are going to verify their account process.
A good example is:
Build a Simple Phone Verification with Twilio, PHP, MySQL, and jQuery[
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