logistic regression - why exponent (log ratio) is linear
p is a probability so it is strictly between 0 and 1. So ln(p/(1-p)) is:

for p = 0: ln(0/1) = -Inf

for p = 1: ln(1/0) = +Inf

So now you've rescaled the probability to +/- Inf. In the GLM framework you can use practically any function that scales probability to +/-Inf (see any GLM textbook, or https://stats.stackexchange.com/questions/20523/difference-between-logit-and-probit-models).

How did it come about? Well, mathematicians and statisticians realised they needed a function with the above properties, had a think, came up with a few, decided on the ones that had tractable asymptotic properties, explored how they worked with real data and decided the sensible ones were logit (as above), probit (see references) and a few others. Of course you should always test your model assumptions against your data, and choice of link is just another one of those assumptions, just like assuming linearity in a covariate.