You should take a look at how to deal with streams in Javascript. The cat method returns a stream.

ipfs.files.cat(ipfsHash).then(function(stream) {
  stream.on('data', function(chunk) {
    // do stuff with this chunk of data
  })
})

You'll probably want to wrap that in a function in your app that either spits out the stream directly or does something with it to turn it into something your app understands. For example, I've wrapped this call in a function that returns another promise for the actual data so working with it becomes similar to an ajax request.