It says that local ecosystems are known to go through critical transitions, and that what's make these transitions remarkable is that the tipping point is so hard to identify or predict. They are extrapolating this and saying that a global tipping point may exist as well. That's it.
You are right, it's incredibly vague and there isn't any awesome undeniable evidence. That's the whole point of the paper. "Hey guys, we might be screwing up our shit more than we think, we need to start taking more measurements" is basically the entire gist of the paper.