Latent Semantic Indexing is a complex indexing and retrieval method which uses a mathematical technique called Singular Value Decomposition (SVD) to identify patterns of word relationships (or in simplified terms- it allows computers to learn synonyms and topic themes in text). While as of writing this particular article, no hard evidence exists that the search engine uses an exact replica of LSI, tests have proven that at minimum, a very similar mechanism is at play.
How this effects SEO is simple. The old SEO methodology of spamming the same word in the same context again and again in an unnatural fashion, in order to increase Keyword Density, is no longer good enough. Today, good SEO copy must be keyword-diverse in addition to being keyword-dense, resulting in a much more enjoyable user experience. Additionally, SEO's undertaking Link Building practices which do not vary Anchor Text, will likely be identified quickly as unnatural, and their efforts devalued.