Johnny Firecloud
Radiohead recorded it normally, then learned to play it backwards. Then they recorded it backwards, and then played that recording in reverse. So you're hearing a reverse recording of the band playing the song backwards.
From CitizenInsane
"Like Spinning Plates" was born out of frustration over an electronic version of "I Will" from the Kid A sessions… Not knowing what to do with [the "dodgy Kraftwerk version"], someone had the idea to reverse it and played it backwards… The reversed backing with its eerie pulse was apparently way more exciting than the ill-fated forward recording, and Thom was also hearing a melody in parts of his reversed vocal of "I Will", which inspired him to write something he could sing over this recording. He went to his notebooks, pieced a set of lyrics together (perhaps using the "cut up lines pulled out of a top hat" method again), thereby creating a new song, "Like Spinning Plates".
However, just singing the new melody and lyrics straight over the pulsing backing track was not good enough. Something different was needed. Perhaps someone had the idea to reflect the fact, that this song came out of turning something backwards, in the singing as well. How about having a backwards vocal track that sounds as if it was sung forwards?
Taken from Reddit: So basically, the band flipped the electronic version of "I Will" backwards an found it to be more interesting, so Thom wrote new lyrics and learned to sing them backwards so that when the vocal track was also flipped, it would have an otherworldly, alien quality while still being sung in English.