Size consistency in computational chemisty methods
Size consistency is essentially the idea that if you were to determine the energy of two noninteracting systems A and B, you should obtain the same result if you sum the energy of A and B separately and if you calculate the energy of A and B as one supersystem (usually with A and B separated by a large enough distance where they don't interact). Expressed mathematically, we might write: $$E(A+B)=E(A)+E(B)$$ provided that A and B don't interact.
As to the second part of your question, I want to address a finer point. Full CI is size consistent; the issue comes when you have to truncate CI to make it computationally feasible. Truncated CI is only exactly size consistent up to the point of truncation. For example, CISD (CI with single and double excitations) would only be exactly size consistent for systems with $N=2$ electrons.
The issue becomes a problem for large systems because this lack of size consistency leads to the wrong limiting behavior of the correlation energy. Specifically, truncated CI leads to the false conclusion that $\lim_{N\rightarrow \infty}\frac{E_{corr}}{N}=0$, while the correlation energy should vary linearly with $N$.
For more information on this, I would read chapter 4, section 6 of Szabo and Ostlunds Modern Quantum Chemistry. The chapter as a whole gives a thorough primer on CI, with this particular section discussing the size inconsistency problem.