The subject of this exercise is the arpeggiated version of the Melodic Minor Scale starting on it's 4th scale degree (Lydian Dominant), over a minor ii-V7-i root movement.
A wealth of information on Melodic Minor harmony can be found in previous posts on this blog, together with ii-V7 exercises.
This scale goes by the name of C Lydian Dominant, which happens to be the mode built from the 4th scale step of G Melodic Minor (Jazz Minor). If we play this arpeggio over an E bass note, we get the sound and function of an E-7b5, the ii chord in the key of D minor.
The rule to remember here is: All Melodic Minor roots and modes are interchangeable with each other.
In measure #1, you're simply playing the sound of G Melodic Minor over E.
The exact same treatment applies for measure #2.
The arpeggio, in this case Eb7#11 13, moves up a minor 3rd from measure #1, while the bass note moves up a perfect fourth or down a perfect fifth to A, forming an A7alt (V7 chord).
The notes of this arpeggio create an Eb Lydian Dominant scale
which is the 4th mode of Bb Melodic Minor. We are, therefore, hearing the sound of Bb Melodic Minor over A (aka: A altered, A altered dominant or A half tone / whole tone).
A common device, effective when used sparingly, is to play a Melodic Minor phrase on the ii chord, and repeat that phrase exactly on the V chord, transposed up a minor 3rd.
This is what's been done here fin the first 2 bars.
Measure #3 (i chord) is made up of an A Pentatonic b6, (A-B-C#-E-F, derived from scale steps 5-6-7-2-b3 of D Melodic Minor). Mucho info on Penta b6 is available on these blog pages, as well as in the available books.
Measure #4 continues the D minor, tonic i chord resolution.
Why was the 7#11 13 arpeggio chosen for this exercise?
Because............It sounds good; and `as the Duke of Ellington once proclaimed: "If it sounds good, it is good!"
Diggeth I do, your Dukeness!