LSI (Latent Semantic Indexing) keywords are keywords that are commonly found together within a single topic and are semantically related to each other.
Let's say your article topic is "horticulture". Typically you would find multiple related keywords like "agriculture", "crops", "botany" and "plants". As your write naturally about a certain topic that you have done enough research for, a certain number of common keyword phrases will be found for that specific topic. These phrases are called LSI keywords - these phrases are semantically linked to each other based on the topic (or the seed keyword) and search engines expect to find them in every article on that topic.
LSI keywords therefore help search engines figure out the main topic of your article. As an example, say your article topic is "cars". If your article had words "clutch, "gas", "price", "mileage", "aftermarket", "used", then the search engines know that your article is about vehicles for transportation. However, if your article had words like "Disney", "animated", "characters", "McQueen" or "mater", the search engines would know that the article is about the Disney movie "Cars". Hence, having related keywords in your content is critical to send the right signals about your topic to the search engines.
Long tail keywords are phrases derived from a seed keyword that consist of more words than the seed keyword does and have lower volume. Examples of long tail keywords for our example seed keyword "horticulture" are "jobs in horticulture", "horticulture therapy", "horticulture online degree" and so on. As you can see each of these phrases has the seed keyword in it, along with a few other words. Long tail keywords are important because they are very specific leading to better conversion rates and have lower volume, which makes them easier to rank for. When you write content you want the content to be focused on a single long tail keyword.
LSI keywords, on the other hand, are not used as the main keyword to focus content on. They are keywords that should also be mentioned in the article to let the Search engines know that the article is truly about the long tail keyword you have focused on. LSI keywords, therefore, work together to help improve the article to rank for the long tail keyword.
Without a sufficient number of LSI keywords, your content will not read naturally. Many content creators tend to stuff the content with the main keyword and ignore LSI keywords. This leads to content that does not look natural. LSI makes it easy for search engines to figure out how natural a piece of content is based on whether enough LSI keywords show up. It has been speculated that Google's Panda update uses LSI keywords to figure out the quality of the content. If not enough LSI keywords are present, the content looks unnatural and hence of poor quality.
Adding LSI keywords to your content makes it look natural. If you are writing in your native language and are an expert in the field you write about, you won't need to find LSI keywords - your content will automatically have them all. However, in cases where you are not an expert or writing in your second language, knowing the LSI keywords will help you write more naturally. The idea is not to stuff LSI keywords in your content - but to know what LSI keywords are expected and see why your content does not have them. You could be using incorrect words or ignoring a certain sub-topic.
The Wikipedia article on Latent Semantic Analysis is a good dive into the technical aspects of LSI. For those who are even more mathematically inclined, check out Stanford's Introduction to Information Retrieval For those who prefer video, check out Standford's NLP course Quora also has a few good questions on this topic that can be found here and here