How we obtain this equation from newton's second law

Just recall Newton's law of motion:

Recall    S =  (v + u)*t / 2. (1)
Similarly    S will be replaced with θ,
                     v  will be replaced with ω
                      u will be replaced with ω0

Therefore θ  =  (ω + ω0)*t / 2.   (2)

Recall from linear motion  v = u + at.
For angular motion it would be  ω = ω0 + αt (3)

The linear acceleration a, is replaced with angular acceleration α

Substituting (3) into (2).

θ = ( ω + ω0) * t/2
θ = (ω0 + αt + ω0) *t / 2
θ = (2ω0 + αt) * t/2
θ = (2ω0t + αt^2) /2

θ = (ω0t  +  1/2 * αt^2).      (Q.E.D)