Can someone explain this like I'm five? If I watch this for 24 hours straight, I would see 24 * 60 = 1440 transitions, would I not? So why film the extra 171?

it is 1611 if you count transitions of individual digits (i.e.: 8:59 -> 9:00 is three transitions because 3 digits changes)

I personally consider it to be 1440 transitions with some transitions involving multiple digits (and hence harder to do) but arguments can be made for both ways of counting.