Storing information as a triple (subject, predicate, object) is a way of representing knowledge. It closely follows the structure of human language and also has the benefit of being machine readable. In the future, this may allow things like AI to take advantage of knowledge that has been stored in this fashion.
Obviously, a triple store is a database that stores triples, and RDF (the Resource Description Framework) is an open standard maintained by the W3C so...
An RDF triplestore has the advantages of allowing for the storage of knowledge in a semantic way, that brings with it a powerful open standard for representing and querying the data.
I haven't used one personally, but I have some exposure to other graph systems like neo4j and redis graph. My understanding is that rdf triple stores are another way to store and describe graph structures. Like the link description says, it's good at modeling social networks, financial transactions, and other 'connected node' based things.
This link compares it to the labeled property graph that neo4j uses if youre familiar with that as a reference. https://neo4j.com/blog/rdf-triple-store-vs-labeled-property-...
If youre not familiar with graph databases and their uses at all tho, I highly recommend a little YouTube exploration on the topic. There's several conference presentations/sales pitches that get into the benefits and use cases for graphs in general. I just got exposed to it this year and it's been super interesting.
For instance, here is what Wikipedia has to say about Triplestore:
https://en.wikipedia.org/wiki/Triplestore
And here is DBPedia representation of it:
http://dbpedia.org/page/Triplestore
And here's how you'd get the data from DBPedia programmatically using SPARQL -
select * where { <http://dbpedia.org/resource/Triplestore> ?p ?o. }
Try it here - http://dbpedia.org/sparql/