Please note: Atheism does not require one to not believe in near-death or out-of-body experiences. It is just rare for atheists to believe in them, so this article will run on the assumption that they don't.
First, we should distinguish between the phenomenon of near-death experiences, and the claim that one's essence/soul/spirit has literally "left the body". Atheists are fine with people experiencing something, but our point of contention is the idea that a soul exists (which is also not an atheist requirement), and that it leaves the body.
Here are the reasons we don't believe in them. Specifically, we'll talk about this in terms of the asserted claims that people coming out of NDEs had access to special information, or saw things, that they shouldn't have been able to see - like information about an ancestor, or seeing an event hundreds of miles away.
This is not meant to be a comprehensive debunking of NDEs/OBEs, but rather an overview of major points describing where we are coming from, and how we approach the question.
Lack of ControlsThe vast majority of the accounts for NDEs/OBEs are anecdotal in nature. They're self-reported from different families and people, who are not in scientifically controlled situations.
We have no means to reliably confirm what exact information was relayed by the experiencing person... nor do we have any means to control for the possibility that someone was talking around the patient, while they were in a semi-conscious state of sleep (we are not dead when we are asleep), while recovering, and then recited the information back. All we have is the assurance of those involved.
For example, if a bother was in a car wreck at the same time, the person may already know what the car looks like, and knew enough about the brother to know what the likely scenarios for accidents are... and if someone merely mentioned the person got in an accident, there's a chance the patient could imagine/dream what happened, and then remembered that imagined scenario later... and if it is interpreted loosely and vaguely enough, could seemingly match - souls not required.
There's commonly a conflict of interest, as well. These reports tend to come from religious people, eager to validate and affirm their religious beliefs, and we have no idea whether they've embellished the report, or succumb to imagination inflation. We don't know whether the family group came to a consensus agreement as to what happened, that does not match what actually happened, without realizing it.
(Yes, we realize we could suffer from a conflict of interest too, to invalidate such claims to affirm our own world views. We are only human, after all. That is why we appeal to standard epistemologies, logic and evidence. That is how one determines what's true.)
Overall, there's a plethora of different ways that the reports can be in error, that we can't rule out. Science has a concept of "double-blind" and "triple-blind" trials, and control groups, that help isolate exactly what's happening, minimizing the risk of contamination of information.
In properly scientifically controlled contexts, where the assumptions and methodology of the studies pass peer review, the data would be more compelling, in terms validating specifics of the phenomenon.
Availability BiasA frequent argument as to the validity of NDEs/OBEs is that they are "very consistent" in the results. That, too, is disputable.
Suppose that each year we had actual 10,000 cases of NDEs (reported or not), where the person would come out of the experience with special information. Now suppose that only one of those was accurate, and the other 9,999 presented information that was inaccurate and wrong.
The 9,999 failures are not going to make the news. They're probably going to fall into obscurity, without anyone knowing or remembering them, before too long.
By no fault of your own, on the receiving-end of the news, you're only going to be aware of the 1 case that had correct special information. From your perspective, all you keep hearing about is that 100% of NDEs reported have bizarre "miraculous" claims like that... when in reality, the accuracy rate is 0.01%.
This is an example of "availability bias", where the information you are receiving is skewed and filtered, causing your understanding of reality to be misled. 
Again, science has methods for controlling this - painstakingly recording and measuring all data to help ensure that the true statistical rates are represented. Being caught "cooking the books" can be career-ending within science.
In Violation of Known RealityIn order for some type of soul or spirit to leave the body, that would require a "brainless mind" - a mind that operates without any kind of biological or mechanical brain.
We have zero examples of this in reality. It is completely unprecedented, and thus, an absurd notion. It is like asking for an running computer program that does not run on a computer.
Further, how is this mind gathering information about the world? Where its eyes and ears, that have retinas and ocular nerves to pick up sight and sound? How would this mind be anything but deaf and blind?
In order for the soul to be real, we'd have to throw out many scientific laws of biology, physics and chemistry.
The Wrong Type of EvidenceOften, people making these claims don't understand how to make a case. The evidence they cite may not even exclusively demonstrate what they are asserting.
If someone "saw" an event that occurred hundreds of miles away, is that evidence for the mind leaving the body, or is it evidence that the mind, while sitting in the body, has X-ray vision with a range of hundreds of miles?
If someone was supposedly talking to an ancestor, who revealed key information that was not known to the person before, does that mean the mind left the body and traveled through time, or that telepathy from a parallel dimension is real?
Different well thought-out experiments, and different types of evidence are needed to pin down exactly what's going on, instead of just trying to confirm what the believers already believe is true (confirmation bias).
Such considerations are applied basics of critical thinking and rudimentary epistemology... which we don't find from the NDE/OBE supporters, often.
ConclusionTo us atheists, the answer seems very straight forward.
A claim that violates known reality, makes claims that require absurd assumptions, and has better explanations in terms of different biases, priming and reporting errors (all of which are heavily precedent and studied), is an easy one to assess.
The claim that a soul leaves the body, or otherwise transcends the body, is undemonstrated and unsupported.