Queen’s random walk (Queen’s random walk). A queen can move any number of squares horizontally,
vertically, or diagonally. Let Xn be the sequence of squares that results if we
pick one of queen’s legal moves at random. 
$(a)$ Find the stationary distribution
$(b)$ Find the expected number of moves to return to corner $(1,1)$ when we start there .
So the answer is $\sum_{x∈S} deg(x)$$ = 1452$, and for the corner $deg(x) = 21$. expected number of moves to return to the corner $≈ 69.14.$
But there are no steps to the answer. I really appreciate if you could show me how to get to the answer, thanks!

The stationary distribution is defined as the normalized number of moves from a given position. In symbols, for a given position $x $ it is $\frac {deg (x)}{\sum_{x∈S} deg(x)}$, where $deg $ indicates the number of possible moves from $x $. 
For a queen on a chessboard, if it is on any of the $28$ squares adjacent to the outer edge (including corners), there are $21$ possible moves ($7$ ranks, $7$ files and $7$ diagonals). If it is on any of the $20$ squares that are in the second concentric frame (i.e., all squares separated from the outer edge by one square), there are $23$ possible moves (because there are two additional diagonal moves). 
If it is on any of the $12$ squares that are in the third concentric frame (i.e., all squares separated from the outer edge by two squares), there are $25$ possible moves (because there are two further additional diagonal moves). Lastly, if the queen is on one of the 4 central squares, there are $27$ possible moves (again two further additional diagonal moves). 
So we have for the corner $deg (x)=21$ and for the total  chessboard $$\sum_{x∈S} deg(x)= 28 \cdot 21  + 20 \cdot 23 +  12 \cdot 25 +   4 \cdot 27 = 1456$$
which leads to an expected number of moves of $1456 /21\approx 69.3$ to return to the corner. 
Note that, in my opinion, the values of $1452$ (instead of $1456$) and the resulting  $69.14$ (instead of $69.3$), both provided in the solutions that you cite, might be the result of a typo (the value of $1456$ is well established for problems on Queen random walks).
