This quotation from the theologian Alister McGrath is a variant of the Dostoevsky quote translated as "Without God all things are permitted".
McGrath focuses on the effect of atheism on morality or Virtue, Dostoevsky's applicability would embrace the othe transcendental Goods of Beauty and Truth - in the sense that without God then there is no final restraint on either dishonesty (lies, hype, spin, propaganda) or beauty (ugliness, horror, banality).
The evidence that McGrath and Dostoevsky are correct is quite simple:
The Twentieth Century.
The 20th century witnessed several enormous atheist political regimes in The Soviet Union, Germany and China - where the implementations of evil were of a scale and thoroughness and of a duration never before seen in human history.
In other words, the removal of God can be seen to have eliminated the final restraint on human brutality.
Of course this is not sufficient 'proof' for those who deeply wish to deny the link; but that is the case for all forms of evidence for anything and without any exceptions.
But to focus exclusively upon ethical aspects of religiously-unrestrained immorality is to miss the fullness of catastrophe which atheism has visited upon humanity - by removal of final restraint because these societies were equally societies of unprecedented institutional dishonesty and ugliness.
In sum, atheism enables the denial of natural law - of all spontaneous ('natural') human acknowledgments of truth, beauty and virtue - and denial implies inversion.
Atheism enables denial of natural law - it does not compel this denial, but it enables the denial to happen when this is expedient.
Because humans always take sides, always exhibit a prefernce, are unable to be neutral.
So when natural law is denied primacy, it is not merely ignored, but reversed.
So we get societies - such as our own - that celebrate the destruction of good, for its own sake; societies that actively will evil - and this is the particular horror of the twentieth century into the twenty-first.
The particular horror of the twenty-first century is that because we are still atheists (indeed, even more so) we have learned nothing from the twentieth century (or, mislearned irrelevant lessons), and have gone a long way towards replicating its specifically modern evils; 'restrained' from doing so only by what appears to atheist modernity as irrational, unenlightened residual conditioning from the bad-old-days.
So the only things keeping modern societies from un-restrained evil are precisely those things which modernity regards as most dangerously evil; and which it is striving so zealously to eliminate.
And it is clear that elimination of 'restraint' is not the worst of things; the worst is that, without God, restraint inverts into its opposite: coercive advocacy of that which was previously restrained.