Feynman diagrams are a series expansion, not unlike Taylor series. We do it for things which cannot be computed exactly. (I don't want to go if it is more "Nature", "our description of Nature" or "a box of numerical tricks we have mastered"... and frankly, I don't believe there is a strong distinction between these three.)

3 lines stand for the lowest non-trivial order (like 'x' in a Taylor series; '1' would be a line) - more complicated things can be made out of these.