When do you just say "He plays" instead of 'He plays computer games", "He plays tennis"?
When you want to express the general action as a whole and expressing detail is unimportant or undesired, you don't use an object.
Person1: "What a cute baby! Is he fussy?"
Mother: "No not at all! He plays, he eats, and he sleeps!"
Another example (It speaks!):
Scientist #1: "Let me show you my new invention."
Scientist #2: "Ok."
Scientist #1 pushes a button. The machine emits some warbling sounds: "wrrbl wrrbl hmm grrr... bbuble..smthing...who?...how?...bababa..baby.."
Scientist #2: "It speaks!" <--------- (As opposed to, "It speaks English!")
Scientist #1: "Yes, it does."
As you already mentioned, this kind of construction most often occurs with intransitive verbs. For example:
He is dining.
She is sleeping.
"To dine" and "to sleep" are both intransitive, so a subject and verb are enough to create a grammatical sentence.
This can also happen with transitive verbs, but usually when referring to non-specific activity.
The child is playing.
Because of what we know about children, we assume the child is playing with toys, playing in the playground, etc., but we don't know the details. Subject-verb is enough here.
The musician is playing.
Because of what we know about musicians, we assume the musician is playing an instrument. We don't know if she's practicing by herself or giving a recital.
But because these examples are transitive verbs, we can be as vague or as specific as we like.
The child is playing football.
Now we know what the child is playing, but we don't know if he's playing with friends or with a team.
So subject-verb sentences are either due to intransitive verbs, or non-specific uses of transitive verbs.