Socrates did not claim to know better than others. But, his point of view was that if a person claims knowledge of a topic or subject, then he must be a master of that particular topic or subject or must have an in-depth knowledge of the topic or subject. Socrates believed that it is better to have honest ignorance than self-deceptive ignorance. Socrates said,
Well, although I do not suppose that either of us knows anything really beautiful and good, I am better off than he is — for he knows nothing, and thinks that he knows. (“Socrates Apology” by Plato)
By applying his Socratic Method, Socrates concluded that experts are ignorant (or unaware) of what things really are. Socrates may also not know the ultimate answers to the questions he asked. But, he knew that things are not as, as people (or experts) are explaining. Thus, he considered himself better than others.