First, I'll preface this by saying that most people, even younger speakers, don't have vowels this progressive in the CVS. Obviously, it's exaggerated for comical purposes, but it's entirely based upon a real phenomenon. For many younger speakers, the vowels are indeed approaching these realizations. The changes, as so often happens, are being led by young females but males are picking up on it, too (this is what happened with the Northern Cities Vowel Shift).
Quotes from the video:
"What the hal?" /ɛ/ -> /æ/
"Shut up, deck, I'm gonna betchslap, you, shetbaag." /ɪ/ -> /ɛ/ and /æ/ -> /a/
"Let's gat em!" /ɛ/ -> /æ/
"Um, I don't think you're gonna fet, I mean, your feet're...kinda beg." /ɪ/ -> /ɛ/
Having lived in both urban Southern and Northern California, I've heard this vowel shift in the under-35 crowd's speech commonly (though I have met a few people in their 40s with such vowels...early adopters? Or just in tune with how the whippersnappers speak?) in all areas of the state, which isn't terribly surprising given the degree of mobility between the two major urban cores of the state (the San Francisco-San Jose Bay Area and the Los Angeles-Orange County-Inland Empire-San Diego conurbation). It's still not entirely clear how or why it originated, but linguist Penelope Eckert from Stanford (in NorCal) has been doing research on it amongst NorCal residents for several years now (in fact, on her page she calls it the "Northern California Shift" though I definitely heard it living in SoCal, too).
But before the "Shoes" video, I wasn't familiar with it being used in pop culture so that's been interesting. It's still a pretty new (though radical) shift but as it spreads knowledge of its characteristics as a regional accent is likely to spread (just as with what happened in the NCVS---50 years ago before it'd majorly spread in the Northern Cities other Americans didn't generally conceive of the Northern Cities as having particularly remarkable accents, though now such dialects are not uncommonly imitated, exaggerated or accurately, in pop culture).