All flavors of Exponential Smoothing (Holt-Winters is one special case) work in the same way. You work with fixed smoothing parameters (typically denoted with Greek letters, $\alpha$, $\beta$ and $\gamma$) and different components (e.g., a level components $\ell$, a trend component $b$ and a set of seasonal components $s$).
How you initially set these values is not fixed, there are various methods. You could fix the smoothing parameters irrespective of the data, or estimate them, and/or estimate the initial component values, e.g., using MLE or minimizing some sum of squares.
In any case, once you have the smoothing parameters and the initial components, you walk through your time series and update the components based on the actual observations. Each Exponential Smoothing model will have its own set of updating formulas. On the FPP2 page you linked, it's right under the heading "Holt-Winters' additive method". You first update $\ell_t$ based on the actual observation $y_t$, the relevant seasonal index $s_{t-m}$ from one seasonal cycle back, and the previous level $\ell_{t-1}$ and the previous trend $\b_{t-1}$. Then you update the trend component $b_t$ and finally the seasonal component $s_t$.
When you have thus walked through your time series, you have these component time series. Table 73 actually goes through exactly this exercise. Try recreating it yourself, taking the first four rows as given (these have been estimated by minimizing the RMSE), and starting the updating at the row for 2005Q1.
If you then want a deseasonalized series, you just subtract $s_t$ from $y_t$.
