[Second major edit since first posting:]
This is a supplement to @CarlMummert's answer, with some slight disagreement or clarification. (He surely knows more than I do in this area, though, so I may have a misunderstanding.)
In the prefix-free machine that D&H describe, the internal structure of the machine is such that it will simply stop reading when it gets to the end of the prefix-free input. Whatever is beyond that point will never be read under any condition. Anything can be on that further part of the tape, and it will make no difference to the behavior of the machine.
One might look at it this way: What sense is there to saying that a sequence of values is "accepted" by the machine when they can never make any difference to the output?
(Or if we do start by thinking that the part of the tape beyond the prefix-free input is also "accepted", then since arbitrary extensions of the prefix-free input are accepted in this sense, we might as well factor them out of what counts as the accepted string, and say that only the prefix-free string is accepted.)
I don't think that prefix-free machines have any special connection to parsing distinct words. There is nothing in the description of prefix-free machines that suggests that they are supposed to advance beyond the end of a prefix-free input to look for a subsequent input. That's not how the concept of a Turing machine is typically used; Turing machines just map single inputs to single outputs (even if they code ntuples).
In another text in the same area as D&H's, Li & Vitanyi 2008 3d, p. 7 (section 1.2) explains that one can encode a pair of natural numbers $\langle x, y \rangle$ as a unique natural number $y + (x + y + 1)(x + y)/2$. In chapter 3, L&V describe prefix-free machines in several ways, specifying in a few places (pp. 200, 201, 209) that when a prefix-free machine reads a program string, all operations halt once the last element of the prefix-free string has been read, and the next element is never read. On page 202, L&V define a universal prefix machine $U$ whose argument three other arguments using the page 7 encoding (cf. p. 105). (On the other hand, later on the same page, L&V define the prefix complexity of a pair of strings using a machine $V$ that reads two prefix-free strings separately and then outputs an encoded version of a pair of outputs. That is like what I take Carl Mummert to describe, but as I interpret the text, $V$ is not a prefix-free machine. It's certainly not one that stops after it reads a complete prefix-free string.)
(I'm not sure whether the read-only forward-only behavior that D&H describe in the quoted passage is required for prefix-free machines. This is part of a question I asked in cs.SE.)