It's fancy wording with some complicated math behind it, but Latent Semantic Indexing is basically a way to find "hidden" relationships between words in order to better understand information.

So-called LSI keywords are more or less just terms related to what you're writing about. The logic here is that if you're writing an article about driving a stick shift car, and included words such as "gear" and "clutch", Google would better uinderstand the context than if they weren't there, and possibly even expect to see these, resulting in either better or worse ranks. 

Sadly, most so-called LSI keyword tools fail miserably at finding actual LSI keywords and most of the creators of these tools even seem to confuse LSI with long-tail keywords, which are completely different things. A lot of both good and bad SEOs hold very strong opinions on this topic as well, ranging from "LSIs are the shit" to "LSIs are bullshit". In my experience, the truth is somewhere in the middle. Whether Google uses LSI or not (which we can't know for sure), it does appear that adding certain terms in your content can be beneficial in terms of ranking better.