Upon selecting a layer to swipe, these two icons appear, and they are clickable, but clicking them doesn't seem to change the swipe action or results. Documentation doesn't mention these icons either.
They are just standard icons in QGIS' message bar. The first is a countdown timer, showing disappearing ticks until the message automatically disappears. The second indicates the severity of the message. In this case, it's just an information message. If it was a warning or critical error the icon would be different.