I am not competent to review this book from a philosophical point of view. Instead, I have tried to give an overview of Popper's themes which might perhaps encourage you to read the book for yourself!
To understand Popper's viewpoint one must understand his case against historicism, for it is in historicism that he sees the roots of the totalitarianism he condemns in Plato, Hegel, and Marx. By "historicism" Popper means the belief that "a truly scientific or philosophical attitude towards politics, and a deeper understanding of social life in general, must be based upon a contemplation and interpretation of human history." Through such study the historicist hopes to be able to identify the "laws of historical development", and so will be able to predict future developments.
Popper's attack on historicism is based on his philosophical analysis of its methods - whether it can live up to its claims to predict the course of social developments - and on the association he finds between historicism and totalitarianism. The philosophical arguments are only sketched briefly in this book: they are developed much more fully in his book "The Poverty of Historicism." In his Introduction Popper writes
Elsewhere, in "The Poverty of Historicism," I have tried to argue against these claims [to have discovered laws of history which enable prophesy of the course of historical events], and to show that ... they are based on a gross misunderstanding of the method of scientific prediction and historical prophecy. While engaged in the systematic analysis and criticism of the claims of historicism, I also tried to collect some material to illustrate its development. The notes collected for that purpose became the basis for this book.
Instead of historicism, Popper advocates a piecemeal approach to social engineering. The question we should ask is not "Who should rule?", but "How can we build political institutions which defend the individual against threats to his freedom?". The future of our world is in our hands, not subject to inexorable historical trends. We may, indeed we will, make mistakes, and not all our efforts to improve the world will succeed, but we should learn from our mistakes. If we turn away from taking responsibility for our actions, and place our trust in historical inevitability, we will only revert to tyranny and totalitarianism.
In choosing his targets, Popper did not make a bid for popularity. However, he demonstrates convincingly that it does not take much more than a simple scratching of the surface to reveal Plato as an advocate of full-blown totalitarianism, including recommending that the State should not stop short of lying to its citizens and resorting to murder in order to keep the élite rulers permanently in power.
Hegel is treated with open contempt. In Addenda I, "Facts, Standards and Truth", written in 1961, Popper writes
I am still surprised that serious philosophers were offended by my admittedly partly playful attack upon a philosophy which I am still unable to take seriously ... [and] ... which I can only regard with a mixture of contempt and horror.
Some more quotes will give a flavour
Hegel's fame was made by those who prefer a quick initiation into the deeper secrets of this world to the laborious technicalities of a science which, after all, may only disappoint them by its lack of power to unveil all mysteries.
Hegelianism is ... exactly what this kind of popular superstition supposes philosophy to be. It knows all about everything. It has a ready answer to every question. And indeed, who can be sure that the answer is not true?
Hegel, installed from above, by the powers that be, as the certified Great Philosopher, was a flat-headed, insipid, nauseating, illiterate charlatan, who reached the pinnacle of audacity in scribbling together and dishing up the craziest mystifying nonsense.
This last quote is not Popper, but Popper quoting Schopenhauer.
Hegel was first and foremost the philosopher of the (totalitarian) Prussian monarchy appointed by Frederick William III in the period after the French Revolution when ideas of "liberty, fraternity, equality" were threatening to take hold in other parts of Europe. Little wonder then that his philosophy is one of the supreme power of the State, and the identity of power and right. By applying his dialectic method he was able to show that the Prussian monarchy was the supreme and ideal form of government that would maintain and grow its strength and dominance through war.
Popper's third target is Marx. He deals far more courteously with him than with Plato and Hegel, admitting he finds him a "far more worthy opponent." In fact there are many parts of Marx' ideas which Popper is in sympathy with, saying that his (Marx') intentions were basically humanitarian. His criticism is of the historicist roots he finds in Marx' philosophy - ultimately in Marx' claim, like others before him, to have discovered the Laws of History, in this case the class struggle. Marx' predictions of the forthcoming social revolution leading to the dictatorship of the proletariat and the eventual withering away of the State have (not surprisingly, Popper would say) proved false. But he gives Marx credit for his perceptive analysis of social conditions in industrial societies in his time, and the realisation that economic factors play a highly significant part in social and political developments.
Popper wrote this book in the early years of the second world war, but in his later additions he stresses that his opposition to totalitarianism is much wider-ranging than the immediate problems of those years. Although the book is over 50 years old, the influence of anti-democratic and anti-humanitarian forces has not diminished, and the force of his arguments is no less.
It would be interesting to speculate what other targets he might have chosen over the second half of this century. Hegelianism is still strongly represented in European philosophy, and it is not surprising that the empty and meaningless twaddle purveyed by Derrida and the deconstructionist movement, which condemned a whole generation of students and academics to at best an indulgence in hollow verbiage, and at worst to an upwelling of bigotry and intolerance, openly acknowledges its roots in Hegel's writings. I can only regret that Popper was not around to turn his attention to this and other threats to the Open Society.