These were all created using the "OOPS" method, in which you take one channel of a stereo song, invert its waveform, and play it back against itself. Because of the way sound waves work, this "cancelling out" can reveal previously buried or mixed down parts of songs. Often it merely creates a crude karaoke version that completely strips out the bass and drums, but for the RHCP, it frequently highlights John or Josh's backing vocals, and there are some other interesting moments. Even in the case where a certain part wasn't exactly hidden, such as John's backing vocals in Savior, by OOPSing the track, you can remove other aspects of the song and hear those vocals a lot more clearly.

Listen here and follow along below:

Blood Sugar Sex Magik
Under the Bridge: Organ and choir clearer at end
One Hot Minute
Deep Kick: Dave's guitar mess during break
Falling Into Grace: Clearer backing vocals, Dave's hidden acoustic, and a weird hum during the chant.
Around the World: Synth during final chorus
Otherside: John's backing vocals
Easily: Guitar "orchestra" at the end clearer
This Velvet Glove: Piano during chorus
Savior: John's backing vocals
Right On Time: John's backing vocals
By The Way
Universally Speaking: Strings, piano, backing vocals (and organ?)
This Is The Place: John's backing vocals
Dosed: Piano
The Zephyr Song: John's backing vocals
Can't Stop: John's backing vocals
Tear: John's backing vocals
On Mercury: John's backing vocals and ending guitar
Minor Thing: John's backing vocals and synth (and amazing sounding drums!)
Venice Queen: John's backing vocals and tambourine
Stadium Arcadium
Charlie: John's backing vocals
Warlocks: Clavinet line a lot louder
Wet Sand: John's backing vocals
Tell Me Baby: John's backing vocals
Hard to Concentrate: Clearer guitar line
21st Century: Guitar at end
Animal Bar: John's backing vocals and fun vocal effects
Storm In A Teacup: Piano and backing vocals
We Believe: Children's choir
I'm With You
Monarchy of Roses: Josh's backing vocals
Factory of Faith: Anthony's "factually, I" effect and Josh's backing vocals
Ethiopia: Josh's backing vocals and piano
Dance Dance Dance: Josh's backing vocals
The Getaway
The Getaway: Josh's backing vocals
Dark Necessities: Clap and piano isolation, Josh's backing vocals
We Turn Red: Josh's backing vocals
Goodbye Angels: Piano during chorus, synth during verse
Sick Love: Percussion during verses
Feasting on the Flowers: Bizarre piano during break
Encore: Strings

Note: this technique also works wonders with bands that recorded their music in a much looser fashion, such as the Beatles