You could use either "a lei" or "le" (indirect object pronoun) with this sentence. They are both grammatically correct. "A lei" emphasizes more strongly that it is her who likes them. When you translate it back to Italian, both are accepted. This sentence is introduced before the section with indirect object pronouns, though, so the default is "a lei."
Already answered just above. (Reading the other comments first will often answer your question). Piacere means 'to be pleasing' So here it says 'to her the short skirts are pleasing' or 'The short skirts are pleasing to her'. So since the skirts are the thing that is doing the acting, the pleasing, the conjugation is with them, rather than with her. And since the short skirts are plural it's piacciono.
For more on piacere and other verbs that act like it see http://italian.about.com/od/verbs/a/italian-verb-piacere.htm
No, perhaps you should have read the comments. http://italian.about.com/od/verbs/a/italian-verb-piacere.htm piacere does not mean 'to like' it means 'to be pleasing', so "To her is pleasing the short skirts", the closest equivalent in colloquial English being 'she likes short skirts.'
Already covered extensively in the comments, but in brief, it's because the skirts (plural) are the things doing the pleasing so the conjugation is with them and thus piacciono - they are pleasing. If you actually read the rest of the comments you will find links to pages explaining it more fully. Advice for the future, always read the thread first, it's a good bet your question has already been asked and answered if it's a fairly long thread.
Literally: Short Skirts ARE PLEASING to her. SKIRTS is PLURAL. Thus, the 3rd person plural of piacere is indicated/required. In conversation, if you made a mistake and said piace, the listener would get the gist of your meaning but how forgiving of your grammar they would be remains to be seen. THE SKIRTS (THEY/piacciono) are pleasing to her. http://www.wordreference.com/conj/itverbs.aspx?v=piacere
Because skirts are plural and piacere conjugates with the thing that is doing the pleasing, not the person who is being pleased. http://italian.about.com/od/verbs/a/italian-verb-piacere.htm