When a customer makes a booking with you using the booking page, you will automatically receive an instant email and SMS (if enabled) notification about the booking.
SMS notification will work only if SMS is enabled on your account. Here you can check if SMS is enabled on your account.
Go to Settings > Settings > Notifications and edit SMS settings.
SMS notifications are saved during "Sleep hours" which are between 11 pm to 8 am (in your own timezone) and sent when the sleep hours are over.