The half step that defines it
Phrygian is natural minor with its 2nd degree flattened. In plain minor the 2nd is a whole step above the tonic, a neutral, stepwise move. Phrygian pulls it down to a half step, so the very first interval of the scale is the tightest one there is, the 2nd leaning hard into the root. Everything dark about the mode comes from that one move.
On the piano it is the white keys from E to E: the 3rd mode of the major scale, C major's notes centered on E. The half step between E and F, sitting right at the bottom of the scale, is the flat 2nd. It is also why the step pattern, uniquely, begins with a half step: H W W W H W W.
On the staff
The Spanish and the heavy
Two traditions live on this sound. Flamenco is built on Phrygian harmony: its cadences descend toward a tonic with that flat 2nd leaning in from above, and the result is the sound most listeners label "Spanish." Much flamenco actually raises the 3rd to major, the related Phrygian dominant scale, but the dark, half-step-above-tonic frame underneath is pure Phrygian.
The other home is metal. The flat 2nd makes a menacing, tense riff almost for free, so Phrygian and Phrygian dominant turn up constantly in heavy music. In both cases the cue your ear is hunting for is the same: a minor scale whose second note sits a half step above the root rather than a whole step. That is the whole identification.
Jazz puts the mode to a less obvious use. Mark Levine notes that players usually take Phrygian not over a minor 7th chord but over a sus♭9 chord, where the mode's flat 2nd becomes the chord's defining flat 9th.
How Aubel handles it
Phrygian unlocks eighth, the fourth of the modes and the last of the minor-family ones. By now your ear is fluent in the minor 3rd, so the question Phrygian poses is narrow: where is the 2nd? Dorian raised the 6th; Phrygian lowers the 2nd. Listen to the very start of the scale. If the second note crowds the tonic, you have it. Rounds play the scale ascending and descending from a random root, so the cue you train is that low 2nd itself.