The paper Early adaptation to eolian sand dunes by basal amniotes is documented in two Pennsylvanian Grand Canyon trackways [1]:
> The narrow width of both trackways indicates that both trackmakers had relatively small femoral abduction angles and correspondingly relatively erect postures. They represent the earliest known occurrence of dunefield-dwelling amniotes―either basal reptiles or basal synapsids...
This geological formation was from the Moscovian age (315.2 to 307 Ma) or the Kasimovian age (307 to 303.7 Ma) in the Pennsylvanian period.
The authors of the paper actually describe their hypothesis for how it occurred [0]. Basically perfect conditions were required to preserve them.
In short the first tracks were made in dry sand, eroded a little, moistened by fog, dew, or light rain, and then covered by dry sand again. The second tracks were made through the dry sand into the moist layer which was then crusted over with calcite and covered. Only the toe markings from the second tracks were preserved.
The presence of a layer composed of a compressed calcium based salt suggests earth that eventually became covered by ocean. Calcium deposits make for fantastic preservatives.
It might be something as silly as change of wind that brought dust instead of sand or maybe different type of sand.
Basically anything that covers the fresh print with a layer that is different enough chemically or physically that will then cause the layers to develop/preserve a boundary that can then break clean off millions of years later after everything is pressed/mineralized/whatever other processes happened.
> among the oldest tracks on Earth of shelled-egg-laying animals, such as reptiles, and the earliest evidence of vertebrate animals walking in sand dunes
In this case they are estimating the ages based upon the known ages of the geologic formations surrounding the layers of sand the tracks were found in [0][1].
There is an entire branch of geology dedicated to answering that question called stratigraphy [0]. It's hard to pinpoint exact dates but we can get a pretty good estimate (313 million +/- half a million is pretty good).
The authors describe how they determined the date in their paper [1]. In this case they know the relative age of the formation the tracks were found in (Manakacha Formation) because we know the age of the formations around it. The ages of the surrounding formations are estimated by the known ages of the fauna (other fossilized organisms) found in those formations (319 Ma below and 312 Ma above).
"The intelligent layman has long suspected circular reasoning in the use of rocks to date fossils and fossils to date rocks. The geologist has never bothered to think of a good reply, feeling the explanations are not worth the trouble as long as the work brings results (American Journal of Science 276:51)." –J. E. O’Rourke
"As long as this cognitive process is acknowledged as the pragmatic basis of stratigraphy, both local and global sections can be treated as chronologies without reproach." (American Journal of Science 276:55). –J. E. O’Rourke
The paper you cited actually goes on to show why the type of dating used is as accurate as it can be. Index fossil ages have been used to calibrate radiometric dating and vice versa. Not all rock formations have radioactive elements we can use to give rocks an "absolute age" meaning relative aging often is as good as we can get.
Yes, carbon-14 is only useful for "recent" samples. The next section of the article discuss other radioactive materials that have a longer half life, and are useful longer ranges.