Aural skills essentially amount to "what theory sounds like." Ideally you want to be able to recognize/imagine the sound of any interval over any harmony. No interval can ever be farther than a tritone from the root (since a perfect fifth above the root is the same as a fourth below the root). So there are only 11 sounds you have to familiarize yourself with. This skill can be expanded to any number of simultaneous notes over a given root (triads, chords, upper structures), or the sound of any note against a static structure (ex// the sound of Db against Cmaj7 - Cmaj7 is the "harmony" and Db is a "melody" note). The important point here is having the root as a focal point for your hearing. It is much easier to hear a large group of notes in relation to the root than to one another.
There are two aural skills: melody and harmony. Melody is the intervallic distance from note to note, and doesn't take harmony into account. Harmony is the way each individual note sounds against the root note or chord.
This approach lends itself better to "outside playing" than thinking of Cmaj7 as C Ionian or C lydian plus some chromatic notes. If you're going to do side-slipping, by thinking of theory this way and the sound this way, you still have context for everything. Instead of playing "Db stuff" over "C" you're just playing over C, and choosing intervals that have a certain desired effect. While initially this may be harder to do, it will result in the player having a much stronger command of the effect they're trying to create.
An effective way to develop this skill is to create a sound loop that you can test different sounds over. Start with just a single root note, and learn how each note in the chromatic scale sounds against it. You want to get to the point where you can jump to any note randomly at will. ***This is much easier to do by recalling the sound of the desired note against the root than it is to think of the interval from one note to the next. This same process can then be applied to a recorded loop of a triad, 7th chord, or any other structure you want to practice with. The goal should also be the same: be able to play any random note of the chromatic scale at will. Eventually this will blur the concept of right and wrong notes, and you will be really free to play whatever you want, because you can hear every single possibility. (If you were a painter, would you not want to have access to every color, even the ones you don't like as much?).
The next step of course, is to get the sounds onto your instrument, and that's what The Third Element of Music: Instrumental Technique will cover.