Why is the set ${1,x^2,x^3...}$ linearly independent as functions? I know that the set is a basis for the vector space of polynomial over some fields. 
but why are they linearly independent?
namely, as function from the polynomial to the base field, what is a proof that they cannot be linearly dependent as functions?

You are having trouble proving this because it's not true. Over the $p$ element field  the polynomial $x^p -x$ is identically $0$ as a function, so $\{x, x^p\}$ is a dependent set.
Those polynomials are in fact dependent over any finite field, since 
there are only finitely many functions from a set to itself and the list of formal powers $x^n$ is infinite.
Note: here is an answer to the narrower question suggested in the comments. In characteristic $p$ the polynomials $1, x, x^2, \ldots, x^{p-1}$ are independent as functions. If some linear combination produced the $0$ function that linear combination would be a polynomial of degree at most $p-1$ with at least $p$ roots (since thare are at least $p$ elements in the field) hence the $0$ polynomial.
