I think we can assume that Otto is not himself Italian, but he likes the Italians and, seeing a desperate situation in which he can help, he comes quickly [hastens] 'to the rescue' [to help]. It's unlikely that the word 'gay' had its current usage when this was written, so 'gay and beautiful people' means lively and attractive people.

Otto, who shared his people's affection for the blue skies and the gay and beautiful people of the Italian peninsula, hastened to the rescue. Otto, who has in common with his own people in their love FOR the blue skies and the happy/lively and beautiful people of Italy , rushed to the rescue. Shared > The meaning here is to " jointly like" > We share the same hobbies/ I share his interest in cars > to jointly use >I share the same house with him / I share the house with him > to give a portion of something to someone > We share the cake equally between 2 of us (50% each) For > Has many meanings and usage . You have to check the word online . The meaning here shows who is getting your affection > I my love for her cannot be put into words. Blues skies of the Italian peninsula > good climate / sunny days

Q: What's the meaning of "share...for..."in this sentence?