"You are mistaken to believe that, if I refute you with the greatest possible effort, I do so for a reason different from that for which I would subject my own words to examination, for fear that, without realizing it, I might believe I know something that, in reality, I am ignorant of; and even now I reiterate that I want to do this, that is, examine the problem, because it interests me and, perhaps, other friends; or do you not think it is a good thing for all men that the reality of every thing becomes clear to them?" [Plato - Charmides]
It means that he is not questioning what his interlocutor says to want to be the center of attention or to prove himself superior to them; he would ask himself the same questions and even contest his own ideas if it served to draw a more precise conclusion. (In essence, the more one seeks to refute, the deeper one goes into the details of a problem's analysis, which is why he would even do so with himself). Because he believes it is interesting for everyone to know the reality (understood as truth, functioning, purpose) of everything.
This quote is practically a hymn to human knowledge, to culture, which tend toward infinity, just as he wants to yearn for this pure and infinite knowledge, even while being aware that one can never know the reality of everything.
