From Wikipedia's article on US Route 24:
When the U.S. Numbered Highway System was started in 1926, US 24 in Colorado was designated US 40S. It began in Grand Junction and went east along the current I-70 corridor to Minturn, from which it followed the current route to Limon. From Limon east to the Kansas border, the current US 24 was designated US 40N. US 40S west of Limon and US 40N east of Limon received the US 24 designation in 1936, when US 24 was extended west from Kansas City, Missouri. The segment between Grand Junction and Minturn was decommissioned in 1975.
In addition, the AAroads website notes that
U.S. 24 historically ran west alongside U.S. 6 from Dowd [close to Minturn – ed.] to U.S. 50 at Grand Junction. The long concurrency was eliminated with approval by the American Association of State Highway and Transportation Officials (AASHTO) on November 15, 1975.
So when US-24 was originally designated in 1936, it ran concurrently with US-6 from Grand Junction to Minturn (143 miles away). In 1975, this westernmost segment of the highway was "de-designated" from US-24. It seems likely that the mile markers were simply left as they were when this was done, resulting in US-24 "starting" at mile 143.
Finally, a scan taken from a 1975 edition of the Rand-McNally Road Atlas:
This appears to show US-24 entering Grand Junction but not leaving it, suggesting that this was the terminus of US-24 at the time.