The literal meaning is "By me there is a cat". The preposition у indicates something nearby, e.g., у окна = by the window. Every preposition requires the accompanying noun to have a definite case, and it just happens that nouns following у must be in the genitive case. Don't try to reason out "why" you need the genitive after у. As von Neumann would say, don't try to understand it; just get used to it. So we use меня and окна after у since they are the genitive form of я and окно.
There is no need to change the form of есть in this construction since it means "there is" and that's exactly what you're saying.
The construction у + genitive есть may seem like a complicated way to express possession if you compare it to English, but you shouldn't expect different languages to convey the same idea in the same way. There are other means of conveying possession in Russian with a grammatical construction that is closer to the English form (e.g., я имею право... = I have the/a right...), but the у + gen. construction is the standard idiomatic method in ordinary circumstances. The usage of я имею is rare in everyday speech. See the question at The usage of иметь.