I don't understand here what does the book mean by expanding in terms of $\delta{q}$ and $\delta{\dot{q}}$ can someone explain that part. I don't understand how did he get that final step?
The change in $S$ when $q$ is replaced by $q + \delta q$ is $$\int_{t_1}^{t_2} L(q + \delta q , \dot{q} + \delta \dot{q}, t) dt - \int_{t_1}^{t_2} L(q, \dot{q}, t) dt.$$ When this difference is expanded in powers of $\delta q$ and $\delta \dot{q}$ in the integrand, the leading terms are of the first order. The necessary condition for $S$ to have a minimum is that these terms (called the first variation, or simply the variation, of the integral) should be zero. Thus the principle of least action may be written in the form $$\delta S = \delta \int_{t_1}^{t_2} L(q, \dot{q}, t) dt = 0$$
The author means "write out a power series expansion (e.g. Taylor series) for the difference $L(q+\delta q,\dot{q}+\delta\dot{q},t)-L(q,\dot{q},t)$. The terms with $q$ or $\dot q$ likely cancel in this difference, so that the power series is written in terms of $\delta q$ and $\delta\dot{q}$, though without knowing what Lagrangian you are using, it is difficult to say for certain that the $q$ and $\dot{q}$ terms cancel.