In his diary, Kierkegaard asks, "Why did Socrates compare himself to a gadfly?" A popular and amusing answer is that it is because Socrates knew that he was a pain in the ass to those whose cherished beliefs and claims of knowledge he kept questioning. Kierkegaard's answer is not as amusing, but much more insightful :
Because he only wished his influence to be ethical. He did not want to be an admired genius, standing apart from the rest, whereby he actually would have made life easier for them, as they would say: "Yes, it's all very fine for him, he is a genius." No, he merely did what every human being can do; he comprehend what every human being can comprehend. There in lies the epigrammatic. He grabbed hold of the individual and worried him, ceaselessly compelling and teasing him with this ordinary, universally human stuff. That made him a gadfly stirring up a person's passion, and he did not permit that person indolently and effeminately to admire and go on admiring, but claimed his very soul. When a human being possesses ethical strength, people like to elevate him into a genius, just to be rid of him; for his life constitutes a claim, a demand, on them. (The Diary of Soren Kierkegaard, edited by Peter Rohde, Carol Publishing Group, 1993, p.123)
The ethical, like the aesthetic and the religious, is for Kierkegaard not just a matter of thinking and understanding, but essentially a matter of living. Living, not just thinking how to live. What Kierkegaard means by "ethical strength" here is thus the strength to live the way one understands to be the right way to live. One's influence would be ethical in this sense if one could affect others in such a way that they not only think about how to live but also live accordingly. And the influence would be doubly ethical if it is by means of exemplifying a way of life that one affects how others live their lives.
Kierkegaard says Socrates "wished his influence to be ethical"; the word "wished" is telling. Not "wanted", not "hoped", but "wished", which suggests that Socrates realized that his influence failed to be ethical (in the above sense). Although he had shown his ethical strength, stirred up people's passion, compelled them to think and examine their own lives, most of them did not change for the better --- did not change their way of life --- as a result. Change is hard; change for the better is even harder.
A person's way of life won't be changed by what he considers to be another person's wisdom, no matter how much he appreciates the wisdom and admires the person who has it. In order for the wisdom to change his way of life, it has to become his wisdom. That is, he has to change first --- the wisdom has to become part of his most natural way of seeing things, part of his worldview, and part of what he identifies himself with. Until then, the wisdom is mere words.
This might be why Socrates denied that he was wise. He didn't want people to see him as wise because he didn't want them to disregard the demands his life made on them as something that only a "genius" like Socrates himself would be able to meet. Examining one's own life and living accordingly is "ordinary, universally human stuff". You can admire Socrates as much as you can, but not even Socrates could examine your life for you. You have to do it yourself, and this is "what every human being can do". But it is hard. Most people would choose to remain no more than an admirer of Socrates rather than do what Socrates did.