Each card shows the city's local time, today's date there, and its UTC offset right now. The offset is the useful part: it is the number you add to UTC to get local time, and it changes when a place switches to or from daylight saving time. London shows UTC+00:00 in winter and UTC+01:00 in summer; Tokyo shows UTC+09:00 all year.
Two cities can show different dates at the same moment. When it is Sunday evening in New York it is already Monday morning in Sydney, so check the date line on each card before you plan a call. The UTC clock at the top is the common reference: if you are agreeing a time with someone in another country, quoting it in UTC removes any doubt.
The clock uses your device's own clock and your browser's copy of the IANA time zone database, so it does not need to be told when daylight saving starts or ends. It only knows what your device knows: a wrong device clock means wrong times here too.