Made in celebration of Miku's birthday, this is a rather chaotic and addictive alternative rock composition which reached over a million hits with incredible speed (less than 32 days which is a feat that most other well-known songs would take from a few months to a year to achieve).
The lyrics are almost nonsensically vague, giving it a crazy and psychotic feeling to the song. Due to this, it can be interpreted in various ways, which has spawned many plausible but not entirely official readings.
Hachi's songs are known to be confusing, deep and open for interpretation. Matryoshka could be literally about people trying to interpret Hachi's song. It's possible he just wrote up a song for the sake of deliberately tempting us, which sounds like trolling, to have a go at deciphering it. With 524, It skips 1 and 3. The numbers 1 and 3 can be read in Japanese as "imi” which translates to “meaning/purpose” in English. Since they are missing, there's "no meaning/purpose."
One interpretation is that in middle of all that madness, there is a hidden inner pain or sadness as is hinted in several occasions in the song, and the madness is just a sort of facade or defence mechanism to bear that pain.
Another interpretation given to the song is that Gumi and Miku are breaking down from the stress put on them by their master(s), who is technically making them sing and dance so much just for entertainment.
The intermissions are descriptions in many different languages from the Wikipedia article for Matryoshka dolls. The first is taken exclusively from the Japanese one, but the second combines information from other languages.