前前前世 is a playful, unforgettable expression meaning "the life before the previous life before the previous life," or more naturally, "three lifetimes ago." It combines 前世 (previous life) with repeated 前 to create the song's distinctive idea of searching for someone across countless reincarnations, making it both poetic and uniquely memorable.
Love that is older than this lifetime. That is the whole idea of this song, and RADWIMPS go all in on it. The title, 前前前世 (zenzenzense), is a made-up stack of 前世 (zense, "past life"). Add two more 前 and you get something like "my past-past-past life". The singer says he started looking for this person that far back: 君の前前前世から僕は 君を探しはじめたよ (I started searching for you from your past-past-past life).
It opens with a funny little scene. He finally finds the person, and instead of a big romantic reunion, they are annoyed: 「遅いよ」と怒る君 (you, angry, saying "you're late"). His excuse is great: 心が身体を追い越してきたんだよ (my heart overtook my body to get here). He was running so hard his feelings arrived before he did.
The scale is ridiculous, on purpose. He came to tell 何億 何光年分の物語 (hundreds of millions, light-years' worth of stories), and they met 銀河何個分かの 果てに (at the end of who knows how many galaxies). Then the cosmic stuff shrinks to one shy question: その手を壊さずに どう握ったならいい? (how do I hold your hand without breaking it?). That swing from galaxies to hand-holding is the best thing about the song.
He is also stubborn. Even if the other person vanished completely, チリヂリ (chirijiri, scattered into pieces), he would また1から探しはじめる (start searching again from 1). Then he tops it: むしろ0から また宇宙をはじめてみようか (or better, shall we start the universe again from 0?). In Japanese, 1から ("from one") is the normal way to say "from scratch", so jumping to 0 and restarting the whole universe is a cheeky joke on that phrase.
A few more lines worth knowing. ぶきっちょな笑い方 (your clumsy way of laughing) uses ぶきっちょ, a casual, affectionate word for "clumsy". 革命前夜 (the night before a revolution) makes the two of them sound like rebels nobody can stop. And 君のハートに旗を立てるよ (I'll plant a flag in your heart) is a bit much, honestly, but it is sweet. The final twist: 君は僕から諦め方を 奪い取ったの (you stole from me the way to give up). He simply cannot quit. Fittingly, this "Movie Version" belongs to the anime film Your Name (Kimi no Na wa), whose story is about two people searching for each other.