Insertion loss equality - proof
As pointed out in PeterK.'s comment, the way it is written this does not make sense. What is meant here is that the insertion loss can be measured on a linear scale (i.e., as a factor) as

$$L(\omega^2)=\frac{1}{|H(j\omega)|^2}$$

or it can be given in decibels:

$$L_{dB}(\omega^2))=10\log_{10}(L(\omega^2))$$

The remaining question could be why $|H(j\omega)|^2$ can be replaced by $H(j\omega)H(-j\omega)$. The reason is the symmetry of the frequency response $H(j\omega)$ for real-valued filters:

$$H(j\omega)=H^*(-j\omega)$$

From this it follows that

$$|H(j\omega)|^2=H(j\omega)H^*(j\omega)=H(j\omega)H(-j\omega)$$