The result is slightly misquoted, it should read $G(X)\leq X^{0.525}$ for $X\geq X_0$. That is, there is no $\log X$ denominator, but $\ll$ can be improved to $\leq$ (when $X$ is sufficiently large). The result appeared in Baker-Harman-Pintz - The difference between consecutive primes, II (Proc. Lond. Math. Soc. 83 (2001), 532-562). If you are interested in the proof, then you need to read it (30 pages).
I recommend you to study the following shorter and less technical paper first: Huxley - On the difference between consecutive primes (Invent. Math. 15 (1972), 164-170). It gives $G(X)\ll_\epsilon X^{7/12+\epsilon}$ for any $\epsilon>0$; in fact it gives much more, namely a short interval version of the prime number theorem (unlike Baker-Harman-Pintz). The prerequisites can be found in Iwaniec-Kowalski's monograph "Analytic number theory".