Spotting it
The reader needs two chords. The first is a major triad or a dominant seventh whose root sits a major third above the tonic of a major key, with at least one note outside that key; the second is the minor chord on the sixth degree. In C major that is E or E7 followed by A minor. The note outside the key is G#, which the fit column names on the first chord's row.
In the table
In C E7 Am F G C the E7's numeral is III7 and the Am after it reads vi. The label sits on the E7, and the span covers both chords, because the rule is about where the first chord goes.
Or a chromatic mediant
The same chord can be two different moves, and what follows it in the line decides which. E major a third above C major, with the tonic's own quality, is the shape of a chromatic mediant, and the source for that move names III as V/vi in so many words. The reader checks the chord after it. When it is the minor vi, the E leans on it and the reader calls it V of vi, and the chromatic mediant rule, which refuses a chord whose root falls a fifth to the next one, stays silent. When the E goes back to the tonic, as in C E C, nothing is being leaned on, V of vi does not fire, and the reader calls the E a chromatic mediant instead. C E7 F C is the same: the E7 goes to F, not A minor.
Major keys only
The rule asks for a major tonic chord and a minor vi. In a minor key the chord a major third above the tonic has a different job, and the rule does not look for it there.