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From javadoc
Initiates an orderly shutdown in which previously submitted tasks are executed, but no new tasks will be accepted. Invocation has no additional effect if already shut down.
That is, calling execute() after shutdown() is an error. Sometimes I consider calling shutdown() a rule. Is it just used to forbid new tasks? Without calling shutdown(), what happens?

Mostly it is used to shutdown the ExecutorService i.e. stop it running when you don't need it.
ExecutorServices contain a pool of threads. Threads are never cleaned up by the GC so you can't just discard them and assume the threads will clean themselves up. Once the threads have been told to stop they will do so after they complete the current task or almost immediately if otherwise idle. After they have stopped they can be cleaned up by the GC
NOTE: Any ThreadLocal values will be retained until the Thread is cleaned up.

The shutdown() method prevents clients to send more work to the executor service i.e no new tasks will be accepted and all the existing tasks will still run to completion unless other actions are taken.Invocation has no additional effect if already shut down

The shutdown method (in short)
starts all tasks, that may be submitted, but not yet started (which could be of type Future<E> for example)
forbids submitting new tasks
makes it possible to gracefully shut down