When an undergraduate in college, I was mainly exposed to so called "Continental" philosophy which seems to have a tendency, to say the least, toward bosh, and to analytic philosophy which seems to have a tendency, to say the least, toward triviality, plus, since it was a Catholic college, selections from Plato, Aristotle, and medieval metaphysicians.
I could not really acclimate myself with these doctrines. It is in a sense unfortunate that I found an alternative outside my formal schooling. With Popper I found someone who is readable--I think that any intelligent general reader can understand him--original, and with an outlook congruent with natural science.
He is known for his ideas on scientific method--that science does not really "prove" theories, but creates conjectures which have rich empirical content and withstand falsification. With ideas like this, decades ago, he attacked doctrines like Marxism, and psycho-analysis. Demolishing the claims of these doctrines may seem to be no big deal today, but decades ago they were major tools of our intellecutal elites.
This book covers old ground such as his views on science but also, it seems to me, breaks new ground. A new contribution is his theory of the Three Worlds, which I think is fruitful. He also deals with the question of free will, which I sense is the weakest part of his book.
Popper distinguishes three realms or 'worlds.' World1 is the world of physical objects; World2 is the world of our subjective beliefs, thoughts, feelings; World3 is what he calls an objective world of knowledge, the objective contents of thought--the knowledge contained in books, musuems, libraries, etc.
Popper holds that most philosophers considered the object of epistemology--the theory of knowledge--to be World2. Popper argues that this is misguided. He thinks it is, to use his words, "irrelevant." The proper object for epistemology is World3. We should concern ourselves not with justifying our subjective beliefs but with objective theories--their contents, the arguments supporting them, etc.
A theory of knowledge based on World3 has some interesting ramifications. It is immune from modern relativistic attacks (this is my personal view). World2 epistemology has premises, both implicit and explicit, that make it vulnerable to relativistic attacks. World3 epistemology, instead, proceeds with the GROWTH of knowledge. Another interesting feature of World3 is that, even though it is man-made, it is autonomous. If humanity were to disappear, World3 will still be "outthere". World3 is created by individuals with certain goals, but the contents of World3 seem to have a life of its own(and this is very metaphorical). It can be used by others in different ways, it leads to new problems and solutions not considered before, etc.
Popper also deals with the problem of understanding in the humanities. There are some who hold that there is a difference between understanding in the natural sciences and understanding in the human sciences--that in fields like history, psychology, sociology, one has to understand by a method which seems to me to be something like a mystical intuitive grasp of the thoughts of another. Popper thinks that this is old hat. The method to, say, reconstruct a damaged ancient text is fundamentally no different from understanding regularities in nature.
Popper died not too long before the advent of the world wide web. It seems to me that Popper's ideas on the three worlds are very applicable to the world wide web. The World Wide Web would fall under the category of World3. It has an ever expanding content of knowledge, of conjectures, of arguments and discussion. Being a part of World3, it is human made, but the world wide web has a certain autonomy. A road built on the web by one person for one thing can be used in different ways by different people.