Tate–Shafarevich group- Solutions in $\Bbb{R}$ and $\Bbb{Q}_p$ but not in $\Bbb{Q}$

Question

Let $K$ be a field. By Wikipedia, “the group measures the extent to which the Hasse principle fails to hold for rational equations with coefficients in the field $K$”. How one can intuitively see this application of the Tate–Shafarevich group?

Moreover, are there any more such important applications to the Tate–Shafarevich group?

Thanks in advance.