Cryptographic Systems Involving Modular Exponentiation
Can someone please clarify the last line for me (in italics)?

Ok, here is a very simple example of what can go wrong.
For a very simple example, let us consider $p=11$ (smaller than what they specify, however it'll act the same, and it'll keep the computations simpler), and $k=3$.
Suppose we want to encrypt the message $m=2$; what we'll do to encrypt is compute $m^k \bmod p$, in this case, $2^3 \bmod 11 = 8 \bmod 11 = 8$.  And, later on in the text, they'll explain how to decrypt.
However, to answer your question, suppose we were to try to encrypt the message $m=13$.  In this case, we'd compute $13^3 \bmod 11 = 2197 \bmod 11 = 8$.  That is, we got the exact same ciphertext that we did for $m=2$; that means that if we get the ciphertext $8$, we wouldn't be able to tell whether the original $m$ was $2$ or $13$ (and by default, we assume that it is less than $p$, namely, $m=2$).
It turns out that this encryption method always handles any $m$ the exact same way as $m+p$; hence if we get a message larger than $p$, there will always be a smaller message that encrypts the same way.
As a side note, when I computed $m^k \bmod p$, I wrote it out as $m^k$ and then applied the $\bmod p$ operation.  This works on these small toy examples, but on realistic cases, we convert the $m^k$ into a series of modular multiplications and squarings, and apply the modulo operation after each multiplication/squaring.  This is not related to your question - I'm just trying to anticipate a possible future source of confusion.