I'm making a guess here, but I think he means specifically the (A) "arrives every weekday at 00:37" part. That has a different implied meaning than (B) "the train arrives every weekday at 24:37." (A) implies that it will arrive every Monday through Friday mornings at 00:37 but will not arrive on Saturday or Sunday mornings at 00:37. (B) implies that it will arrive Tuesday through Saturday mornings at 00:37, but not Sunday or Monday mornings.

In the context of the train departure time, it's easy to work out what is meant by a weekday 00:37 arrival, but if they were divorced from each other for some reason, weekdays at 24:37 would be easier to interpret.

right - 24:37 friday makes is read as "37 past midnight on friday night" - which is friday night as humans consume time, but saturday morning by the calendar.

Being awake past midnight brings up a few challenges with calendars. My watch wil count a late (late) night walk against tomorrow, not against today. As a human, I consider "tomorrow" as "after I've gone to sleep", not "after midnight". 00:37 is still very much tonight.

An app I have to track habits and streaks lets you specify the time to count as "that day". I often walk my dog after midnight, so I have this set to 3am.

Think of it as a 24 hour day that starts at (to use UK public transport's setup) 4:30. You want the timetable and the users to agree on what 8am means, and that 8am and the following 2am are the same work day of 24 hours.