I have 3 answers, depending on what level you're asking:

1. There's no such thing as causation, and maybe not even time and change.  Everything was determined in the initial configuration of quantum waveforms in the distant past of your lightcone.  The experience of time and change is just a side-effect of your embeddedness in this giant static many-dimensional universe.
2. Causation cannot be determined by pure observation.  It can be inferred pretty strongly, if there are "natural experiments" that isolate A and B, and A happens before B and we can't find anything else that might be upstream causes.  Yes, absence of evidence, in the bayesean sense, is evidence of absence.
3. Causation can be pretty strongly determined by directed experiment.  The experimenter IS the upstream cause of the change, and carefully manipulates A in ways that other causal effects are isolated from what happens to B.

Remember, causality (like everything about undersanding the world) is a model - it lives in the map, not the territory.  All models are wrong, some models are useful.  This one turns out VERY useful in making your way through the world, and in understanding likely chains of future events.

To check if A causes B, you can check what happens when you intervene and modify A, and also what happens when you intervene and modify B. That's not always possible though. You can consult "Causality: Models, Reasoning, and Inference" by Pearl for more details.

This is an interesting question even though I'd want to reframe it to answer it. I'd see the question as a reasonable response to the standard refrain in science; "causation does not imply correlation." That is, "well, what does imply causation, huh?" is natural response to that. And here, I think scientists tend reply with either crickets or "you can not prove causation, what are you talking about". 

Those responses seem satisfying. I'm not a scientist through I've "worked in science" occasionally and I have at times tried to come up with a real answer to this "what does prove causation" question. As a first step I'd note that science does not "prove" things but merely finds more and more plausible models. The more substantial answer, however, is that the plausible models are a combination of the existing scientific models and common sense understandings of the world and data. 

A standard (negative) example is the situation where someone found a correlation between stock prices and sunspots. Basically not going to be pursued as a theory or causation because no one has a plausible reason why the two things should be related. Data isn't enough, you need a reason the data matter. This is often also expressed as "extraordinary claims require extraordinary evidence" (which also isn't explained enough as far as I can tell). 

Basically, this is saying natural science's idea of causation rests on one big materialistic model of the world rather than scientists chasing data sets and then finding correlation between them (among other things, the world is full of data and given some data set, if you search far enough, you'll find another one with a spurious correlation to it). Still, the opposite idea, that science  is just about finding data correlation,  is quite common. Classical "logical positivism" is often simplified as this, notably. 

Moreover, this is about the "hard" sciences - physics, chemistry, biology and etc. Experimental psychology is much more about chasing correlated and I'd say that's why much it amounts to bald pseudoscience.