Impact of distinguishing between random text and cipher text? In theory, distinguishing cipher text from random text is considered insecure for any PRP algorithm.  Say for example - due to Patarin's proof with about six rounds of Feistel Network - the attacker can distinguish cipher text from random text after about $2^{(n/3)}$ queries to the encryption oracle, where $n$ is input size. For example, if the input domain is 32 bits, after around 1552 queries the attacker can distinguish cipher and random strings.  

In practice, what does this mean and what exactly is the impact? How will the cipher text look after say 1553 in above example? Will the attacker get to know the key too or will he be able to guess the next few cipher texts?
Patarin's proof isn't about distinguishing ciphertext from "random text", it's about distinguishing a Feistel cipher from a random permutation.  That proof is also in the information theoretic world, which means that the Adversary is computationally unbounded, and as such can deduce everything about the cipher (the 'key', all remaining ciphertexts, etc).

In the more 'practical/useful' sense (i.e. against bounded Adversaries), if the Adversary can efficiently distinguish a cipher from a random permutation then they can usually mount efficient key-recovery attacks against the cipher structure (guessing part of the key in order to 'peel off' rounds, and using the 'wrong-key randomization' hypothesis).  Once you know the key then you can deduce all future ciphertexts.

I should also add that any permutation, whether truly random or pseudorandom, can be distinguished from "random texts" /"random strings" with the following Advantage (probability of success):
$$\mathbf{Adv} \leq \dfrac{n(n-1)}{2^{\ell+1}}$$
Where $n$ is the number of requested chosen plaintexts and $\ell$ is the blocksize of the permutation in bits.  This is because if you select $n$ distinct plaintexts you will always get $n$ distinct ciphertexts if the Oracle is a permutation.  But if the Oracle is returning random strings then there might be a collision in the outputs (and the probability of collision grows with the amount of requested plaintexts, becoming essentially $1$ when around $2^{\ell/2}$ requests have been made).