What makes a sentence?
A sentence needs three things:
A noun (subject)
A verb (action word)
A complete thought
What do sentences have to do with fragments?
Simply put, fragments are incomplete sentences. They are missing either a subject, verb, or a complete thought.
Fragment: All day on the beach.
What is it missing? A subject and a verb. How do we fix it? By adding a subject and a verb.
Complete: We played all day on the beach.
Sometimes sentences have subjects and verbs, but they need a complete thought.
Fragment: After they finished dinner.
It has a subject (they) and a verb (finished), but because of the word "After", we are left wanting to know more. What happened after they finished dinner? We have to complete the thought.
Complete: After they finished dinner, they went to the movies.
For a more detailed explanation, read the following document and watch the following video: