Show that the single-tape TMs that can not write on the portion the portion of the tape containing the input string recognize only regular languages
The general idea is that the TM has only finite memory while being in the input part of the tape. It can do arbitrary computations in the right part of the tape, but it can only observe a finite memory footprint of the input. It starts in the input part (on the left) and it can reenter the input part (from the right) arbitrarily often. But each time it enters the input part, the only thing that matters is the state it has when entering (or the fact that it starts from the left). And the only outcome of processing the (read-only) input part is the state it has when it exists (or the fact that it does not exit because it has accepted or rejected in the mean time).

1) That is already  the reason for the domain and codomain. The domain is everything that matters about entering the input part, the codomain is everything that matters about exiting that part. Hence the function completely describes the behaviour of the TM on the input part. In a sense, the function contains everything the TM will ever find out about the word $s$. 

Both domain and codomain are finite, hence so is the set of functions. Although the definition is about infinitely many words $s$, there will only be a finite number of different functions $F_s$. 

2) $F_{\varepsilon}$ is not about $\varepsilon$-transitions. And it does not expand the previous definition. It is just the special case $s=\varepsilon$, that is the case of the empty input word. Or rather, the case of the empty prefix. After all, the paragraph where this appears is all about how $F_s$ can be computed incrementally along prefixes of $s$. 

3) By now we have fully defined the automaton, except for acceptance. And the automaton state fully describes what the TM can do with the word we have read. We have not yet said anything about the part of the TM's computation that happens outside of the input word. The trick is that we do not have to delve into that computation. We just say that the state $F_s$ is accepting, if the word $s$ belongs to the TM's language. The paragraph has an explanation why that is well-defined.