The particular use case mentioned for LemonGraph is "streaming seed set expansion", which, given a set of "seeds" can expand that set based on their communication patterns to find people that are likely in the same community or overlapping communities.
E.g., if you have a few known terrorists and a database of metadata about phone calls or internet communications (emails etc.), you can analyze who your known subjects (seeds) talk to and who those people talk to, etc., to identify communities. This relies on the fact that there tends to be cross-communication between people in the same community.
This kind of analysis can often reveal the structure of communities, like where the headquarters, who the boss is, etc.
In industry and law enforcement, the same kinds of approaches can be used to identify fraud of various kinds.
"What it's like" is a graph database, in this case based on a fast in-memory database to support high speed graph analysis on a single machine.