According to a scientific study, we are hard wired to accept people's apologies, even if they don't really mean it. As humans began to live in groups for protection, we evolved a complex set of behavioral traits that ensures amicable relations between members. From the Scientific American:
If we were perfectly angelic specimens, we wouldn’t need the sociometer
to begin with; rather, the sociometer is as much a preemptive device
for disarming our selfish desires and preventing dips in our relational
value as it is a corrective one that prompts us to repair the
reputation-related damage we’ve already done. One quick-and-dirty
damage control tactic is apologizing to those we’ve wronged. And you
might be surprised to learn just how effective a simple apology can be.
In fact, a recent series of studies showed that, to a large extent, it
doesn’t even matter if the apology is patently insincere—at least for
the target of the original wrongdoing.
A great example in action: