For any $\epsilon > 0$ and for any finite interval $I \subset [1, \infty)$, there is a $T_{0}$ such that for all $T > T_{0}$ and all $\alpha \in I$ we have $|F(\alpha,T) - 1| \leq \varepsilon$. The meaning of the conjecture is that the Fourier transform $F(\alpha, T)$ of the pair correlation of zeros up to height $T$, of the Riemann zeta-function converges to a limit $F(\alpha)$ as $T$ goes to infinity. You can invert the Fourier transform, and then read off the conjecture as saying that for any smooth Schwartz class function $f$, the limit $$ \lim_{T \rightarrow \infty} \frac{1}{N(T)}\sum_{0 < \gamma, \gamma' < T} f(\gamma - \gamma') $$ exists, where $N(T)$ is the number of zeros up to $T$. Moreover the limit is an explicit linear functional of $f$. It coincides with the similar functional that one gets from considering the pair correlation of the eigenvalues GUE matrices. For better or for worse a lot of ink has been spilled on this last observation.
The importance of the conjecture is that it immediately implies that the normalized gaps between zeros of the Riemann zeta-function tend to get arbitrarily small. With a bit more work this then implies that there are no Siegel zeros. This was in fact Montgomery's original reason for considering the pair correlation of the zeros.
P.S: In your first display you should be dividing by $T \log T$ and not $T / \log T$, since $N(T) \sim (1/2\pi) T \log T$.