React Native version check
npm view react-native version
You can check the latest version of react native. You can run it under the root of your project or everywhere. Of course, you will need internet connection to check latest version.
react-native -v
If you run under the root of project, it will return the version of two npm packages, react-native-cli and react-native. Those versions are using in your project.
Just wanted to clarify Aung's answer:
npm view react-native version
checks globally while react-native -v
ran inside project's root folder checks version of the app itself.