Licensing and useful links
JetBrains Gateway itself is a free application. However, you should have a license for the remote IDE.
The licensing for Remote Development is handled on the local machine and is covered by your existing active IDE license.
No license information is passed to or saved on the remote server. You can start the IDE backend without entering the license information since JetBrains Gateway is a free application.
The license will be checked later when you connect to the launched IDE backend. Product in the license on your local machine must match the remote backend IDE. If you use a corporate floating license server, that's your local (client) machine, which must be able to reach this server. In this context nothing is changed from the same local WebStorm mechanism.
The following links can help you quickly find the needed information:
For the information on how to configure the Remote Development environment and open your project, refer to Connect and work with JetBrains Gateway.