Excess Carrier Generation and Recombination
Recombination is a statistical process that depends on the amount of available holes and electrons. Therefore, you can express the average recombination rate as a product of the two

$$R = \alpha_r\cdot n\cdot p$$

Generation can have a number of sources (eg. excitation by light - photogenerated carriers), and can be pretty much take any form in the equation - (eg. by modulating the light source).

Because in thermal equilibrium we know that the following equation holds:

$$n\cdot p = n_i^2$$

The equation

$$\frac{dn(t)}{dt}=\alpha_r\left[n_i^2 - n(t)p(t)\right]$$

is therefore simply constructed in such a way that the previous equation is always true for whatever \$\alpha_r\$ you fill in. In other words, for thermal equilibrium:

$$\frac{dn(t)}{dt} = 0 \Leftrightarrow n_i^2 = n(t)\cdot p(t)$$