How can science disprove a higher power (or god) in general?
Science can, and only cares to, disprove definite, testable claims. So the question is not IS there a higher power, but how does some person claim to KNOW there is a higher power in ways that are observable (measurable) and then how do those observables demonstrate the reality of said higher power.
In simple terms, the one making a claim that something is real (a positive ontological claim) has the burden of proof.
For example, if the answer is "well, there is a higher power because my prayers are answered". That is a claim that can be tested. But the additional question is, is merely asking for something and then receiving actually evidence for a higher power? When you dig into these claims they usually vanish as being either Confirmation Bias (you ask for 100 things, 1 mundane item comes true - god exists!) or Wishful fulfillment (See Also: But I Had A Personal Experience).
Here it is worthwhile to study things like illusions (how our brains can be fooled) and things like Cognitive Biases.
Consider the alternative epistemic (how you approach knowledge or truth) approach, famously posed as Russell's Teapot.
So, how can science disprove a China Teapot in orbit about the sun?
Sure! I'll send a probe there, give me the coordinates.
Oh no, sorry it's invisible and undetectable (here I have Moved The Goal Post - a tactic often used by theists)
If we cannot DISPROVE the China Teapot must we then accept that it is true? Surely not.
We cannot absolutely disprove the teapot but it's an extremely reasonable position to be aTeapotist -- there almost certainly is NOT a Celestial teapot in orbit about the sun, it is no leap of faith to believe so. So saying "I disbelieve your claim of a Celestial teapot" is therefore a reasonable atheistic position - Even though it allows a (vanishingly small) possibility of being incorrect.
If, on the other hand, you believe a claim to be reasonable, but simply cannot find evidence to prove or disprove the claim then an Agnostic position may be more fitting (and many people will disagree with me on this point - and I admit to an oversimplification). Some might say that the multitude of anecdotal evidence, combined with possibly personal experiences is sufficient that we should remain Agnostic on the question. Some people see that as lending undeserved credibility to the claim.
Which label you use is of little practical difference, what really matters is your position, your evidence, and your reasoning.
I would argue that the idea of god is made-up. I base that on a deep study of ancient History and Shamanism (including my own personal 'Shamanic' experiences). And even though I have personally seen burning books, "angels and demons" (I would say entities), touched a lightray, and many other such things I absolutely do not believe in a spirit, or soul, or god, or higher power, or ghosts, or intelligent causative agency, etc. I believe in the power of my brain to create entire worlds and immersive experiences as it does while dreaming.
By most definitions I am therefore an atheist and I happy take up that position in a discussion on the subject.
But strictly philosophically speaking, I identify most strongly with Thomas Huxley's brand of Agnosticism. But not in the same way that most people use the term 'agnostic'. This is my blog entry on it if you have any interest in the philosophy of it: http://iconoclasm2000.blogspot.com/2010/12/on-agnosticism.html
However you identify (atheist or agnostic or antitheist or antheist or all of the above) I highly recommend reading some of the works of Thomas Huxley (who was also known as Darwin's Bulldog), Bertrand Russell, and Robert G. Ingersoll.
Google Books has a lot of these older texts available for Free! [including Darwin's Origin of Species, which is also available as a free 'amateur recording' audio book: ]