A new patient opens your booking page on the 14th and your schedule is full until the 22nd. Without help they click through days one at a time, hitting No times available, and some give up. The date picker shows where the openings are before they look.
A month of availability
The date picker is a month grid. A dot under a day means it has at least one time the visitor can book for the chosen service and location, and arrows move between months. The calendar's day cells use the location's time zone, so Today is the clinic's today.
Past days and days with no open time can't be selected. Picking a day shows its times beside the grid, or tells the visitor to try another date.
Next available
Next available sits beside the Date and time label. It searches ahead, within how far in advance your booking rules allow, and jumps to the first day with a time. If nothing is open in that window, it says no upcoming availability was found and to check back later.
The same jump works from a link. Add ?date=next to your booking link, with &service=... if you like, and the page opens already on the first open time. It runs once, so a visitor who then picks another day is not pulled back. The embed builder has a matching Open on the next available time option in Embed booking on your site.
What a visitor does next, holding the time while they sign up, is in Reservation holds.