I can't understand where the equation in the blue circle comes from or is derived from, I know it says the sum of the perpendecular components should be zero, but can you explain visually? and what exactly does the B angle, the flow angle, represent? $$-u\sin B+v\sin B=0$$
You have to look at the velocity decomposition at location $y_u$. Below I made a velocity diagram at location $y_u$:
If you decompose $u$ and $v$ into the red velocity (tangential to the surface) and the blue velocity (perpendicular to the surface), which formulae do you get?
Then the text goes on to say that the velocity perpendicular to the airfoil surface is 0, so what happens when we apply that knowledge to our just-derived equations?
That equation just translates the fact that the speed perpendicular to the surface must be zero since the airfoil is "airtight" i.e. no air can flow though the wing. This condition is normally termed "non-penetration".
$\beta$ is simply the slope of the airfoil and equals, by definition, $dy_u/dx$ and $dy_l/dx$.
Being $u sin \beta$ and $v cos \beta$ the speed components perpendicular to the airfoil, the non-penetration condition simply gives that their sum must be zero. Mathematically this is called "boundary condition".