The more correct form is 'There are', because the subject of the sentence is the whole noun phrase 'A lot of cats', which is plural in meaning. Even though the word 'lot' is singular, it has no meaning in itself in this sentence. It is simply part of a multiword phrase meaning 'many'. It's the same as saying
'There are a hundred cats'.
Even though 'a hundred' is singular, the whole noun phrase 'a hundred cats' denotes a plural number of cats, so the verb should be plural.
That said, it really doesn't matter which you use. Both are fine. 'A lot of' is informal, used mainly in spoken language, so there is more tolerance of variations.
It is much easier to say 'There's..' than 'There are..' when speaking, so that's the form that tends to be used more, because it feels more natural. It is even acceptable, in informal contexts, to say 'There's lots of cats', even though it makes no sense grammatically.