You did it correctly, but as the error states, the module needs to be re-compiled (GWT Compiled). Once recompiled, Force your browser to reload (Ctrl + F5). If that doesn't work, you can look at this answer I wrote regarding cleanup of compiled output. Do what's listed there, and try running GWT compile again.
To run production mode outside Eclipse, you must do a GWT compile and create a WAR file, which you can then deploy on a Java application server such as Tomcat. This process is much easier with an ant build script.
To make an ant build script for your project:
You can easily create an ant build script by running the webAppCreator command from the directory where the GWT library is installed. That command will create a sample project just like clicking "New Web Application Project" in Eclipse, except it will also make a build.xml file.
You can run it in Eclipse by right clicking build.xml > Run As... > Ant build. By default it will run the build target. It also has a war target, which you can run to create a WAR file.
For more details on the webAppCreator, look here.