Question

how was the theory of Herbert Spencer used by hitler ?

Answer 1

The idea of Herbet Spencer which was only valid for nature was used by nazi party to prove their ideas of racism. According to Herbet Spencer in nature only the fittest survives . The nazi party who claimed that  pure blood nordic aryan was the most superior and strongest race and jews were the lowest rung and so only aryans should survive.
 

Answer 2

Herbert Spencer later added the idea of survival of fittest. According to this Idea , only those species survived on earth that could adapt themselves to changing climatic conditions.