Two tetrachords, two characters
Split melodic minor in half. The lower four notes (tonic, 2nd, flat 3rd, 4th) are pure minor: a whole step, a half step, a whole step. The upper four (5th, raised 6th, raised 7th, octave) are exactly the top of a major scale: whole, whole, half. So the scale starts out clearly minor and finishes sounding like major. That split personality is the thing to hear, and it is what makes melodic minor distinct from the other two minors, which keep a flat 6th.
Raising the 6th is what separates it from harmonic minor. Harmonic minor left the 6th flat, opening the augmented 2nd up to the raised 7th. Melodic minor pulls the 6th up too, and the gap closes. Every step in the upper half is now a whole step or a half step, nothing wider. You get the leading tone without the exotic leap.
On the staff
The ascending-versus-descending question
Melodic minor is the one scale here with two competing definitions, and it is worth being clear about which one you are training. In the older classical definition the raised 6th and 7th apply only on the way up; on the way down the scale reverts to natural minor, flattening them again. The reasoning is that the raised degrees exist to lean toward the tonic, so they are dropped when the line is descending and that pull is not wanted.
The contemporary jazz definition, the one Aubel uses, keeps both degrees raised in both directions. This version is a scale in its own right, sometimes called jazz minor, and it is the parent of a large family of improvising sounds. Its tonic chord, a minor triad with a major 7th on top, is the so-called "spy chord" you hear under tense film scenes. If you learn the scale as a fixed shape that sounds the same up and down, you are learning the form the app plays.
In modern jazz the scale does more than name a key. Mark Levine builds a whole family of chord-scales from its modes: the fourth mode is the Lydian dominant played over a dominant 7♯11, the sixth is the bright scale for a half-diminished chord, and the seventh, the altered scale, sits over an altered dominant. Learn melodic minor as one shape and you have all of them.
How Aubel handles it
Melodic minor unlocks fourth, right after harmonic minor, the two of them finishing the trio of minor scales before the modes begin. Pairing them in sequence is deliberate: both raise the 7th, so the question your ear has to answer is whether the 6th is raised too. Harmonic minor leaves the gap; melodic minor closes it. Rounds play the scale ascending and descending from a random root in the jazz form, the same both ways, so the cue is the smooth major-flavored top over a minor bottom.