Brownian motion construction on the unit sphere in Oksendal text book

Question

In example 8.5.8 of Oksendal text book, the Brownian motion is constructed on the unit sphere. I don't understand how he defines the time change $Z_t(\omega) = Y_{a(t,\omega)}(\omega)$. He defines the unknows in the next lines as $$ \alpha_t=\beta_t^{-1}\,,\quad \beta_t=\int_0^t\frac{1}{|B_s|^2}\,ds\,, $$ but I cannot understand this notation at all. Also, how can we understand that "it is reasonable to call Z Brownian motion on the unit sphere"? It is not also clear why the function $\phi(B)$ equals Z.