Unwilly,
If N is actually a power, say 13^7, then clearly N will continue to be a power modulo any prime. In this case, N==13^7 mod p. His claim is not extraordinary.
If N is not a power it might become a power modulo p. For example, 2 is not a power, but 2==3^2 mod 7.
Does that answer your question about the highlighted text?
