What is the purpose of decomposing ciphertext into digits during relinearization in Brakerski Vaikuntanathan homomorphic encryption?
The decomposition helps with the noise growth of your scheme.

You see, the decryption only works if you "error" (here the terms you denote by $e$) is small enough. If $t*e$ grows beyong the modulus (all the computations are done modulo some number, let's say $q$), the mod $t$ operation won't yield the message $m$, but $m$ + $t*e$ mod $q$.

Now when you're multiplying 2 ciphertexts, the errors get multiplied, and they also get multiplied by the plaintext.
Hence, you need your plaintext to be small enough, so $s^2$ cannot be used directly for this reason. When you decompose it in binary, you will have a sum of $log(q)$, but each of those will have a much smaller noise, because they only get multiplied by at most $t$.