Those are your (or another creature's) tracks. Red means that they are easy to see, and faded means they are harder to see. The redness is based on both the freshness, and your ability to track others (via the Tracking skill).
Rain and time will cause tracks to disappear gradually, as will the "Hide Tracks" button. Tracks left at the beginning of the turn are considered to be older than tracks left at the end of the turn, hence why they are more visible towards the end of a turn.
Lastly, some creatures will follow tracks in the direction from least to most fresh, so beware if your tracks cross a creature's path.
Hope this helps!
Regarding the footprints disappearing on the next turn, this is a function of the tracks degrading over time. When the turn ends, an hour or so passes, causing all tracks to become more faded.
If the tracks fade below your character's tolerance for detecting tracks, they disappear entirely.
Ideally, the fading would scale linearly to your minimum detection threshold, rather than going from solid to invisible like that. It's possible that the system has a bug, or perhaps the hour is enough to go from fresh to beyond the character's threshold.
In either case, I agree that's a bit unintuitive, so I'll add a to-do to look into it. Thanks!