Renato's explanation is correct, I will just provide a few more examples so you can compare. José me tiene miedo (a mí) = José is scared of me. José te tiene miedo (a ti) = José is scared of you. José le tiene miedo (a ella / a él / a usted) = José is scared of her / him / formal you. José nos tiene miedo (a nosotros) = José is scared of us. José les tiene miedo (a ustedes / a ellos) = José is scared of (plural) you / them. In all the sentences is possible not to say the part in parenthesis, but the objective pronouns (me, te, le, et cetera) must be in the sentence. The second part is optional depending whether the other person knows whom you are referring to or not. For example: José me dijo que no quería esa película, le tiene miedo (a la película). José told me he did not want to see the movie, he is scared of it. In this case "the movie" was just mentioned in the previous sentence, therefore there is no need of mentioning it again in the following one, that is why the objective pronoun "le" is enough to make it clear that it is the movie that he is scared of. But in: Hablé con José hoy, me dijo que le tenía miedo a la película. I spoke with José today, he told me he is afraid of the movie. In this case, the movie was not preivously mentioned, that is why it must be mentioned later on in the second part, and the objective pronoun "le" is not optional it has to be there. You have to take into account that the objective pronouns (me, te, le, et cetera) are always the object of the verb, the information that comes after it is just "more information", that is why you cannot omit the pronouns from the sentence, even if the actual noun is there. I hope this helps you.

"le" is a pronoun. We use it when an object or person is receiving an action from the subject. In your sentence "los gatos" is your object and "ella" is your subject, also only the object modifies it. Él "se" está duchando (a él mismo). // He is taking a shower. Roberto "os" compró este regalo (a ustedes). // Roberto bought this gift to you. Yo "les" pagué el almuerzo (a ellos). // I paid them lunch. Here it is explain better http://roble.pntic.mec.es/msanto1/lengua/1pronomb.htm

I don't know how to explain it. I hope this may help you: http://spanish.about.com/od/pronouns/a/le_with_certain_verbs.htm