Why does the frequency response H(f) exhibit conjugate symmetry in case of a linear system with real impulse response?
Definition of frequency response:

$$H(e^{jw}) = \sum_{k=-\infty}^{\infty} h[k] e^{-jkw}$$

Now take the complex conjugate of both sides.

$$H^{*}(e^{jw}) = \left(\sum_{k=-\infty}^{\infty} h[k] e^{-jkw}\right)^{*}$$

$$H^{*}(e^{jw}) = \sum_{k=-\infty}^{\infty} h[k] \left(e^{-jkw}\right)^{*}$$

$$H^{*}(e^{jw}) = \sum_{k=-\infty}^{\infty} h[k] e^{jkw}$$

$$H^{*}(e^{jw}) = H(e^{-jw})$$

Because h[n] is real, the complex conjugate only applies to the exponent term.