The actual page number will differ depending on the publication; however, this quote usually comes approximately nine pages into Act I of Oscar Wilde's play (and because I am getting my version from an anthology of English Literature, it's on my page 640).  Regardless, this quote comes in the middle of the conversation between Lady Bracknell and Jack ("Ernest") Worthing.  Jack, who calls himself "Ernest" in the city, has just asked Gwendolen to marry him.  Lady Bracknell, who proclaims that Mr. Worthing is not on her very important list of eligible bachelors, is not thrilled at this prospect of marriage. 

Right before the quote you mention, Lady Bracknell becomes glad to hear that Mr. Worthing smokes because "a man should always have an occupation" (Wilde 640).  Going even further, Lady Bracknell is glad to hear that Mr. Worthing is twenty-nine.

Lady Bracknell. A very good age to be married at.  I have always been of opinion that a man who desires to get married should know either everything or nothing.  Which do you know?

Jack (after some hesitation). I know nothing, Lady Bracknell.

Lady Bracknell. I am pleased to hear it.  I do not approve of anything that tampers with natural ignorance.  Ignorance is like a delicate exotic fruit; touch it and the bloom is gone. (640)

Lady Bracknell, then, is the character who says these lines.  Lady Bracknell puts a great stress on this quality of what she calls "natural ignorance."  Here she explains that ignorance is easily taken away because, once one learns anything, ignorance is gone.  Lady Bracknell becomes less and less fond of Jack ("Ernest") Worthing, especially after learning that he has no real relations to speak of.  This is unacceptable for a good, Victorian lady in regards to marriage.