According the instructions for form 8453-OL, you keep the form for your records.  You mail it in if the state request to see your form.

https://www.ftb.ca.gov/forms/2017/17_8453olins.pdf

B Taxpayer Responsibilities Before you e-file your return, you must: • Verify all information on the form FTB 8453-OL, including SSN(s), ITIN(s), and banking information. Reconfirm your routing and account numbers. • Inspect a copy of the return and ensure the information is correct. • Sign form FTB 8453-OL after you prepare the return, but before you transmit it. 

After you e-file your return, you must retain the following documents for the California statute of limitations period: • Form FTB 8453-OL • Original Form(s) W-2, W-2G, and 1099R • A paper copy of Form 540, Form 540 2EZ, Long or Short Form 540NR • A paper copy of your federal tax return • A paper copy of your other state income tax return if you claimed the California Other State Tax Credit. Refer to California Schedule S. The California statute of limitations is the later of four years from the due date of the return or four years from the date the return is filed. (Exception: An extended statute of limitations period may apply for California or federal tax returns that are related to or subject to a federal audit.)