How does the present perfect affect the present?

I have seen in a cartoon when a scientist invented something he exclaimed right away "I did it! I did it!" Why not "I have done it!" - since it affects the present?
If a lion ate my left arm yesterday I cannot say "A lion has eaten my arm yesterday" because of the word "yesterday" :S But it DOES affect the present very much since I do not have an arm. My other question is how to learn this tense when American media keeps using the simple past in place of the present perfect. I know all the rules from the grammar book regarding this tense, yet I cannot feel how people think when they use this tense. My language has only 1 past tense.