A typical workflow for a Palantir customer was that Palantir would come in and dump a ton of data out of old crufty databases and into Palantir's datastore. Then, they'd establish connections between that data. This is all sounds kind of hand-wavy, but the gist of it is that a lot of government agencies have data that lives in separate databases and they can't easily correlate data between those two databases. Once the data was in Palantir's system, they could do queries against all their data, and make connections and correlations that they wouldn't otherwise be able to find when the data was previously siloed.
One of the sample use cases was identifying people filling prescriptions for schedule II drugs multiple times on the same day, and correlating that with pharmacies run by people connected to known drug traffickers. Previously, this was hard to do because the database of prescription purchases was disconnected from the database of drug convictions.