Translation:I want to participate in the Olympics.
You can just think about the word "exit" in English.
A sentence that breaks your thinking of "wo deru" and "ni deru"「村をオリンピックに出ます」- "I exit the village into the Olympics."
Movement verbs in English can already imply more than physical movement. Consider: "I was into fishing." "I will go into the translation industry." "I left my old habits behind." "I approached the problem well prepared."
It's fine to imagine yourself walking towards and away from a concept. You can usually get the meaning right just by having that picture.
It's the difference between the Japanese and English languages. In Japanese the word exit/leave (go out) and showcase are the same word.
Think of a "coo coo" clock and the bird "coming out/going out" of the clock (i.e. Leaving the clock). That's 出.
Now pretend that the "coo coo" bird is an athlete going out onto the field and "bingo-bango!" Now you have our lesson example!
Another example would be a model that's too sexy for his shirt... (You know what I mean?)