It's not an accurate measurement, but given this wasn't a government funded enterprise, I doubt anyone had access to more sophisticated method. And for those without an expensive pocket watch, they simply watched the sun passing by overhead.
Until the Mormons came up with the "roadometer" in 1847, it likely was an amalgam of different averages(and wildly varying degrees of accuracy). They knew roughly what "the average _____" could do in certain circumstances, so based on that(and the matter that caravans move at the speed of the slowest member) they could "make an educated guess" on ground covered in general.
More specifically however, there were "trail guides" with landmarks, and those landmarks had (roughly) known travel distances between them. So while they might not know total distance traveled in a given day. They would know when they passed Chimney Rock and when they reached Independence Rock. At which point they could come up with an "average day's travel" based on the distance traveled and time taken.
A group with an in-person guide rather than a printing of dubious origins would be likely to have other more numerous guide points to use. As to how those distances were determined? Who knows? Might have been some random guy with a sextant and pocket watch, or maybe someone with more formal training in surveying.
More often than not, they probably made a educated guess based on prodigy to various landmarks they were aware of. Obviously, someone with "local knowledge" would be better at such guesswork. Unless they had a roadometer, of course.