How to follow analysis in books?

Set up a real  chess set and board and follow the game on it. It is the way that people have done it for hundreds of years

You could also use the analysis board here. It would have the advantage of letting you go through a variation and then backtrack to where the variation starts. (You need to use the browser version. The apps can't do it.)

"... Just because a book contains lots of information that you don’t know, it doesn’t necessarily mean that it will be extremely helpful in making you better at this point in your chess development. ..." - Dan Heisman (2001)
https://web.archive.org/web/20140626180930/http://www.chesscafe.com/text/heisman06.pdf
"... The books that are most highly thought of are not necessarily the most useful. Go with those that you find to be readable. ..." - GM Nigel Davies (2010)
"... If it’s instruction, you look for an author that addresses players at your level (buying something that’s too advanced won’t help you at all). This means that a classic book that is revered by many people might not be useful for you. ..." - IM Jeremy Silman (2015)
https://www.chess.com/article/view/the-best-chess-books-ever

Yes, set up a board and follow along.

Another possibility is to use a digital board, such as with chess.com or Chessbase or any other computer board.  You can enter the moves, and as these record the notation as well, you can double-check that you are doing it right.  One frustration I had with learning notation, especially descriptive notation, was making a wrong move but only finding out 10 moves later when I reached an impossible position.  Digital boards can help catch this early, especially when you are still getting the hang of notation.

As a final option, many games can be found online or in databases, so if you do a search and load the game directly, you don't need to set up a board or enter moves.  Obscure games won't be found, but most games featured in books can be found this way.  While I prefer using a real board, this undoubtedly saves time, and I've been using it more and more.

If you have enough table space set up two boards and pieces.  The larger board is for the main line or main variation of the game.  The smaller board or side board is for playing through all the side variations.   The side board is also called the analysis board. 

The main board you keep clean and use it to reset your analysis board as you progress through the game.  

As has been pointed out.  Use a real board, and pieces.  You want to simulate OTB playing as much as possible.  Also, its amazing how much more you will see with a real board, and pieces, as opposed to staring at a 2D set.

Note that there's usually an Index of Variations in the back of the book in case you want to explore making a move other than the one you're following.

Note also that some opening books are little more than lists of variations.  I like books that explain - at least through every few moves WHY the move was made in terms of possible tactics and strategies.

Cyrus Lakdawala's series of Move By Move books (Caro-Kann, Slav, etc.) are excellent, often asking "student" questions like, "What is wrong with White's last move?" where the answer is "After the swap, White remains with a Bishop stuck on the same color as his central Pawns."

Consequuently, you may not want to play all of Lakdawala's move choices, but you understand what to avoid!