Where does -1 in BJT’s Ies equation come from?
That is the Schockley Diode Equation, if you want to look it up.  If you're an EE student sit up and pay attention in your solid-state quantum mechanics class -- you'll end up deriving this one there.

The 1 comes from the rate of electron-hole pair generation in the junction; it is constant over voltage (although not over temperature -- I was in the industry for about 5 years before I realized that \$I_{ss}\$ is temperature dependent.  I felt cheated that I was not told sooner).

The \$e^\frac{V_{BE}}{V_T}\$ part is from the rate of recombination of hole-electron pairs; it is exponential with voltage, and exactly matches electron-hole pair generation at \$V_{BE} = 0\$ (as one would hope it would).