The fermi function $f(E)$ gives the probability that a given available electron energy state will be occupied at a given temperature.
The fermi function comes from Fermi-Dirac statistics and has the form
$f(E)=\frac{1}{E^{\left(E-E_{F}\right) / k T}+1}$
The basic nature of this function dictates that at ordinary temperatures, most of the levels up to the fermi level are filled.
At higher temperature, a larger fraction of the electrons can bridge this gap and participate in electrical conduction.
Hence, at finite temperature, the probability decreases exponentially with increasing band gap.