How did Weibull or any other mathematician arrive at the above expression? I saw the 1951 paper, but it is not clear to me. In 1939 he had published a book called "A Statistical Theory of the Strength of Materials", and I am currently searching for this book. In the meantime can anyone explain me.
The paper "Strength of materials and the Weibull distribution" by Eric S. Lindquist in Probabilistic Engineering Mechanics 9 (1994) 191-194 probably has what you want. I found several online copies with a "weibull's original paper" google search, but they might be behind institutional paywalls.
Weibull's 1939 paper "A statistical theory of the strength of materials" seems to have appeared in what WorldCat says was this journal:
Ingeniörsvetenskapsakademiens handlingar = Proceedings / Royal Swedish Academy of engineering sciences the Royal Swedish Institute for scientific engineering research = formerly: Proceedings / the Royal Swedish Institute for engineering research
in volume 151, pp.1-45. I have not found any on-line copies of this paper. WorldCat lists libraries on 3 continents with this journal; the nearset to me is about 100 km away from where I live, the 6-th nearest in another country.
Two of the 3 parameters, $\gamma$ and $\eta$ (this is me talking, not Lindquist) are location and scale parameters, more or less inevitable in any technological application. The third, $\beta$ is the interesting one.