0. Your question suggests (by use of the word "the") that there is only one basis for the natural cubic splines. There are in fact many, and this is only one particular basis for them.
1. It sounds like you want an intuitive way to interpret $d_k(x)$. Nice, intuitive ways of understanding things don't always exist, and I doubt such an interpretation exists for the $d_k$. Hastie, Tibshirani, and Friedman certainly don't mention one. Nonetheless, we can visualize the functions $d_k(x)$, however, which might help. Let's take the knots to be the set $\{ 0, 1/3, 2/3, 1 \}$. Then we can plot the functions $d_k$ and $N_k$, along with an example of a natural cubic spline given by the linear combination of the $N_k$ with weights $(\beta_1, \beta_2, \beta_3, \beta_4) = (1, 10, -50, 100)$.
4. (I know this is out of order) The natural cubic splines differ from cubic splines in that they are linear outside the interval $(\xi_1, \xi_K)$. The functions $x^2$ and $x^3$ can't be part of the basis because they are not linear outside the interval $(\xi_1, \xi_K)$. Any function in the basis has to itself be an element of the function space, and $x^2$ and $x^3$ are not natural cubic splines because they don't satisfy the linearity condition outside $(\xi_1, \xi_K)$.
2. Now, the reason we need to the basis functions $N_k$ is the same reason that $x^2$ and $x^3$ can't be basis functions: every basis function must be linear outside the interval $(\xi_1, \xi_K)$. The $N_k$ are constructed in the post you linked in order to satisfy this constraint. By taking the differences $d_k(x) - d_{K - 1}$, the quadratic and cubic parts of these functions cancel out on the interval $(\xi_K, \infty)$, which makes the resulting functions linear on that interval.
3. Perhaps the authors could have mentioned that $k$ is an index/variable that ranges over the values $k = 1, \dots, K-2$. This means that the expression $N_{k + 2} (x) = d_k (x) - d_{K - 1} (x)$ for $k = 1, \dots, K-2$ represents the $3^{rd}, 4^{th}, \dots, K^{th}$ basis functions.
