What is meant by PCA preserving only large pairwise distances?
Don't confuse PCA with dimensionality reduction.
PCA is a rotation transformation that aligns the data with the axes in such a way that the first dimension has maximum variance, the second maximum variance among the remainder, etc. Rotations preserve pairwise distances.
When you use PCA for dimensionality reduction, you discard dimensions of your rotated data that have the least variance. High variance is achieved when points are spread far from the mean. Low-variance dimensions are those, in which the values are mostly the same, so their absence is presumed to have the least effect on pairwise distances.
I am currently reading up on t-SNE visualization technique and it was mentioned that one of the drawback of using PCA for visualizing high dimension data is that it only preserves large  pairwise distances between the points. Meaning points which are far apart in high dimension would also appear far apart in low dimensions but other than that all other points distances get screwed up. 
Could someone help me understand why is that and what does it mean graphically?. 
Thanks a lot!