In fact, can someone actually satisfactorily explain to me what the fuck angular means in regards to music?
Well, "angular" in the sense I've used it refers to melodies or counter-melodies where notes move contrary to the scalar or arpeggiated models that have been used throughout the history of music. Angular melodies usually tend to focus on odd intervals or scales.
That probably didn't explain it too well, so here are some examples in modern music.
Buck and Judy - Deerhoof
Listen to the guitar part throughout, especially around 2:30.
Atlas - Battles
This song has several instances of well-formed angular melodies riddled with parallelism. Both vocal melodies revolve around contrasting perfect intervals: the "chorus" is comprised solely of cascading perfect 5ths and the "verse" is sung in parallel 4ths. The guitar solo(?) lies heavily on the backbeat, uses major 7ths (rarely used in melodies), and then switches to a flatted 7th mid-line. The entire breakdown section hinges on repeating the 9th tone (2nd tone). I could probably go on and on about this song for ages. In fact, most Battles songs have several angular melodies in them.
If you want examples of classical music that is angular, try
The Rite of Spring by Igor Stravinsky
I wish I could give some more examples right now, but I have class.