The Weierstrass Approximation Theorem states that any continuous function defined on a closed interval can be approximated as closely as desired by a polynomial function. This means that for any continuous function, there exists a polynomial that can get arbitrarily close to the function's values at every point in the interval. This theorem is significant in real analysis and has practical applications in various fields, including numerical analysis and computer graphics. It assures that polynomials, which are simpler to work with, can effectively represent more complex continuous functions within a specified range.