The answer to the question of how much is the Discovery App will depend on which Discovery App is being referred to. The Discovery GO app is essentially free. Users only need to have a cable/TV subscription package that supports Discovery Channels. Check the list of supported TV providers on the official website of Discovery GO. Going to the official website while having a U.S. IP address and providing the cable TV provider login details usually unlocks all the available content for free.
If the Discovery App being referred to is the Discovery Plus app, then the price changes to $6.99 per month with no ads and $4.99 per month with ads. The Discovery Plus app is a completely different streaming service, similar to Amazon Prime Video and Netflix.