Here is a much better explanation of PCA: https://stats.stackexchange.com/questions/2691/making-sense-...

The key insight that many are missing is that PCA solves a series of optimization problems, namely that reconstructing the data from the first k PCs gives the best k-dimensional approximation in terms of the squared error. Even more, this is equivalent to assuming that the data lives in a k-dimensional subspace and becomes truly high-dimensional because of normally distributed noise that spills into every direction (dimension).