People acquire skills in the real world by putting in the effort to practice them to proficiency. Being very intelligent might allow you to attain proficiency with less practice, with might being the operative word. However, intelligence is not going to substantially help you become proficient in physical skills like climbing, leaping, or swimming any more than having a natural talent for athletics is going to help you become proficient at math or foreign languages. Acquiring a skill is far less a function of intelligence than it is a function of determination and time.
As for "what is Intelligence," it is what it is. In the D&D rules "what is intelligence" is defined for us as logic, education, memory, and deductive reasoning. And you are naturally free to adjust that definition as you see fit because the rules are not absolutes.