Can anyone explain the relationship between the refractive index, the speed, wavelength and angle of a wave?
in my book is states that $$n = \frac{v_1}{v_2} = \frac{\sin θ_1}{\sin θ_2} = \frac{λ_1}{λ_2}$$ but it doesn’t state any explanation for the relationship.
Well, if you draw phase fronts at the boundary, you should be able to convince yourself that the transverse component of the wave vector needs to be the same in both media:
$$ k_1\sin\theta_1 = k_2\sin\theta_2 $$
so
$$ \frac{\sin\theta_1}{\sin\theta_2} = \frac{k_2}{k_1} = \frac{\lambda_1}{\lambda_2} $$
(If you want a picture, put it in the question).
The dispersion relation in media is:
$$ \omega = vk = \frac{ck} n $$
Put that in the first formula and you get Snell's Law.
It's just boundary conditions. Phase on each side of the boundary has to match, at all ties.
