It makes more sense when you approach it from linear algebra.

Like you can make any vector in R^3 `<x,y,z>` by adding together a linear combination of ` <1,0,0> `, ` <0,1,0> `, ` <0,0,1> `, turns out you can also do it using `<exp(j2pi0/30), exp(j2pi0/31), exp(j2pi0/32)>`, `<exp(j2pi1/30), exp(j2pi1/31), exp(j2pi1/32)>`, and `<exp(j2pi2/30), exp(j2pi2/31), exp(j2pi2/32)>`.

You can actually do it with a lot of different bases. You just need them to be linearly independent.

For the continuous case, it isn't all that different from how you can use a linear combination of polynomials 1,x,x^2,x^3,... to approximate functions (like Taylor series).