The Gaussian measure is the probability measure on $\mathbb{R}^d$ with density $$\frac{e^{-\|x\|^2/2}}{(2\pi)^{d/2}}.$$
The ball is defined as $$B_r = \{x\in\mathbb{R}^d: \|x\| \le r\}.$$
How to compute $$\mathbb{P}(B_r) = \int_{x\in B_r}\frac{e^{-\|x\|^2/2}}{(2\pi)^{d/2}}dx?$$
The integrand is radially symmetric so we can use spherical coordinates and not have to worry about the angular coordinates. $d^dx\propto \rho^{d-1}d\rho,$ so $$ P(B_r) \propto\int_0^re^{-\rho^2/2}\rho^{d-1}d\rho \propto\int_0^{r^2/2} e^{-u}u^{d/2-1}du = \gamma(d/2,r^2/2)$$ where $\gamma$ is the lower incomplete Gamma function. The proportionality constant can be found by normalization.