Here's how it works: in any given year, a handful of dates land on the same day of the week. That day is what Conway calls the "doomsday." For 2015, the doomsday was Saturday; for 2016, it's Monday. (See the video below for more on how to calculate a year's doomsday). Some of those dates are easy to remember: 4/4, 6/6, 8/8, 10/10, and 12/12 all land on the doomsday. It takes an easy mnemonic device to recall some others: 5/9, 9/5, 7/11, and 11/7 can be remembered with the sentence "I work 9 to 5 at the 7-11." There are also a few one-off doomsday dates: March 14th (3/14 or Pi Day, for those who use the day-month format), the last day in February (regardless of leap year), and January 3rd or 4th, depending on if it's a leap year.
Once you have those dates memorized, you can calculate the weekday of any other date via its proximity to a doomsday. For example, what weekday did Dame Judi Dench's birthday, December 9, land on in 2015? That year's doomsday was a Saturday, so we know that 12/12 landed on a Saturday. The 9th is three days before the 12th, so Judi Dench's 81st birthday landed on a Wednesday. Still confused? Watch the video below to clear it up.