Assuming a server setup that correctly reloads (redeploys the webapp) whenever server-side code (and/or resources) changes (this is outside the scope of GWT; same for debugging).
You run CodeServer with launcherDir pointing to a folder served by the server (or DevMode with -noserver, and -war pointing to that same folder), it then generates a *.nocache.js file in the directory (and copies public resources there too).
When you load your page in the browser, the *.nocache.js is loaded and triggers a compilation in the CodeServer.
Whenever you change client-side code, refresh the page in the browser and it'll trigger a recompilation.
Whenever you change server-side code, redeploy the webapp (depending on the setup, this can be entirely automatic, or involve some manual action).
To debug client-side code, use the browser's devtools, or your usual SDM/IDE integration (I never used any, so can't really comment)
When using my gwt-maven-archetypes for example, "mvn tomcat7:run" will automatically redeploy the webapp whenever resources or classes change in target/classes. Most IDEs will happily compile your code (automatically on save, or triggered by a keyboard shortcut) into target/classes, which would trigger a redeploy. "mvnDebug tomcat7:run" (or running the Maven goal for debug in your IDE; for example, in Eclipse, Debug As… → Maven… → tomcat7:run) allows you to debug your server-side code. And "mvn gwt:codeserver" launches SDM for the client-side code.
AFAIK, the GWT Eclipse Plugin has some "one click" way of running both a server runtime (configured in Eclipse WTP) and SDM, and I'd assume that Eclipse is smart enough to make redeploying after changes either completely automatic or only a keyboard shortcut / click away.
The "connections" you're talking about are the standard JDWP protocol for connecting to a JVM to debug the server-side code in the server runtime, and SDM with SourceMaps (and possibly a "remote debugging" protocol of your browser if you want IDE integration rather than debugging right in your browser).