What you ask is not true, as @TheoJohnson-Freyd explained. What is true is that $I$ is an ideal of a polynomial ring, and $I_0$ is the “initial ideal” of $I$, i.e., the ideal generated by the initial terms of all elements of $I$ (so, a monomial ideal), then the dimension of $V(I_0)$ is the same as that of $V(I)$ (because they have the same quotients as graded vector spaces, so the same Hilbert-Samuel function). The magic property of Gröbner bases is that the initial terms of the basis elements suffice to generate the initial ideal.
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