alphabet of a single tape turing machine that simulates a multitape TM
(I successfully downloaded the image).

It is not clear in the picture what is the difference between $\Sigma$ and $\Sigma_A$, however looking at the definition of Exercise 4.12 I think that the author simply distinguishes the input alphabet $\Sigma_A$ from the working alphabet $\Gamma_A$ (tape alphabet); so in order to simulate:

the input tape of A (with symbols from $\Sigma_A$) plus the $k$ tapes of A (with symbols from $\Gamma_A$) 

it uses $\Sigma_A$ in the first term of the Cartesian product.

Be careful that you didn't copy exactly the $\Gamma_B$ of the book (you wrongly put a $\cup$ where the book puts a $\times$):

$\Gamma_{B} = (\Sigma_{A} \cup \{¢,␣ , \$ \})\times \{\uparrow, ␣ \} \times (\Gamma_{A}\times \{\uparrow, ␣\})^{k} \cup \Sigma_{A} \cup \{¢\}$

The final $\Sigma_A$ is probably used to represent the initial input of the multitape machine A "as-is" (and satisfies the requirement that the input alphabet must be a subset of the working alphabet); but I think it is a little bit confusing (and perhaps the $ symbol should also be included with the final ¢)

The first steps of the simulation of B should be: copy the input to the  "hypotethetical" track 0 of B, initializing the head pointers of the "hypothetical" tracks 1,..,k of B.