In a word, Math.
The power of LSA technology is based on mathematical properties and special computations that can identify the relationships of meaning between all of the items in a collection of information.
The computations behind LSA are fairly complex, but the basic concept looks something like this: Given a collection of items of information it is possible to represent the collection as a simple, but very large, table with a row for each item (article, document, etc.) and a column for each term (word, phrase, etc.) in the collection. The value in each of the cells of the table is simply the number of times the term for that column appears in the item for that row. This table is essentially a numeric matrix. We adjust the numbers in the matrix using various weighting schemes based on the numeric properties of the data in the matrix. The weighted matrix is then processed using a mathematical algorithm to reduce this large matrix to a multi-dimensional hyperspace where each item and each term is represented by a vector projecting into this space. You can roughly picture this with a simple 3-dimensional representation where the vectors point out into the 3D space.
This illustration is an extremely simplified representation using only 3 dimensions so you can visualize it. In practice we use a hyperspace, typically with anywhere from 300 to 500 dimensions or more.
When this processing is completed, information items are left clustered together based on the latent semantic relationships between them. The result of this clustering is that items which are similar in meaning are clustered close to each other in the space and dissimilar items are distant from each other. In many ways, this is how a human brain organizes the information an individual accumulates over a lifetime.
In addition to simply retrieving data items, the entire data collection can be analyzed based on meaning in several different ways. This allows a large information collection to be digested and examined from a number of angles that are simply not possible using other techniques.