Simple subject is the main subject, without the adjective included. Sometimes, there is no adjective, so the complete subject can also be the simple subject.
Example: The talkative girl got an A on her test.
In the sentence above, the complete noun is smart student. It's describing the student as smart, so it is an adjective. The simple subject is student.
Simple predicate is the verb. Simple predicates NEVER have a helping verb in it. The simple predicate usually comes right after the simple subject.
In the same sentence, the simple predicate is got because the verb is got.
Here is the sentence:
The laughing child played on the new swing.
The simple subject is child. The child is the one playing on the new swing.
The simple predicate is played. The verb tells what action the subject did. In this sentence, the subject(child) played on the new swing.
Simple subject: child
Simple predicate: played
child | played ↓
The laughing child | played on the new swing.