Why does bootstrapping (R)LWE homomorphic encryption produce small noise?
The output of bootstrapping has relatively small noise because it starts from an encryption (of the secret key) that has very small noise, and performs some homomorphic operations on it. These operations increase the noise somewhat, but it starts out so small that the result still doesn’t have much noise.
It doesn’t really matter that the ciphertext being bootstrapped has large noise, because that ciphertext only defines the function we (homomorphically) evaluate, namely, decryption of that fixed ciphertext by the (encrypted) secret key. In other words, the secret key is the input to the function we homomorphically evaluate, and it is encrypted with very small noise, so the result has only moderate noise.