Here's why:
* "Like" in this sentence acts as a preposition, similar to "to" or "from".
* "I" is the nominative case of the pronoun, used when the pronoun is the subject of the sentence.
* "Me" is the objective case of the pronoun, used when the pronoun is the object of a verb or preposition.
Since "me" is the object of the preposition "like", it is the correct form to use.