It seemed to me that two recurring themes in Volume 1 of this 3-part anthology were characters with mental powers, and the aftermath of nuclear war. If I had to pick two themes for this volume, I would say they are resistance to, or defiance of, authority and hard-science explorations of alien worlds and alien races. One of the stories has both.
I don't know if the editors of the various volumes are consciously organizing the stories along these lines--their prefatory remarks, at least in these first two volumes, don't contain any indication of that. But the patterns are certainly clear.
In this volume, the stories that have elements of resisting authority in order to do what's right, or what's needed, are "Industrial Revolution," "Say It With Flowers" (probably my favorite story in this entire collection, and one that would make a splendid basis for a feature film treatment), and "Marque and Reprisal," all of which have characters who really know how to think outside the box.
Anderson was known for his creation of alien worlds and races that conformed to known physical and scientific rules, and his penchant is on full display in this volume with stories like "Queen of Air and Darkness," "The Longest Voyage," "The Pirate," "To Build a World," "The Corkscrew of Space" (a fun story set on Mars that shows how necessity can be the mother of invention), "A Little Knowledge" (which has a great plot twist), and the fittingly-named "Epilogue," which is both the last story in the volume and also a haunting farewell to Earth in the far future, that made me feel like I understood how the dinosaurs would feel if they were to see how we exist in the world they ruled for so many eons.
"A World Called Maanerek" falls into both categories, taking place on an alien planet colonized by humans and requiring the protagonists to resolve a clash of loyalties and cultures.
Of course, there are lots of other stories in the mix as well--Anderson was too full of ideas to get pigeon-holed into a couple of story patterns. "Operation Afreet" is a very fun mixture of magic and technology, and you'll have fun reading about the weird combinations he has come up with. There are a couple of time travel stories, "Brave to Be a King" and "My Object All Sublime," which give Anderson an opportunity to show off his masterful knowledge of history.
"Brake" puts its characters in a tight spot after a terrorist incident and they have to think their way out if they are to survive; warning that this one includes some pretty egregious stereotyping of females which was still pretty normal in 1957, when it was written. "The Burning Bridge" features a similar dilemma, though it is a moral one and not an astrophysical one. It has a great line, "No one in his right mind wants to be a colonist." "Innocent at Large" is a light-hearted story that reminds us not to judge books by their covers, while the very clever "Uncleftish Beholding" sees Anderson playing with his knowledge of the English language's Anglo-Saxon progenitors by describing atomic physics in language that a Viking could understand. "The Critique of Impure Reason" was my least favorite work in this collection. It reminded me of some of the stories in Asimov's "Robots" milieu, but I just couldn't identify with the characters, particularly the robot. It did have eerie overtones of today's AIs though, and how they can go off the rails if they are not trained on the right material--very prescient for something written in 1962.
Finally, the volume also contains three essays by Anderson: "Science Fiction and History," which dovetails nicely with his two time-travel stories; "Science and Creation" which gets a little into cosmology and religion, and which I feel didn't need to be written; and my favorite, "The Hardness of Hard Science Fiction," which explains his philosophy of creating alien worlds and races, which many years ago inspired me in the creation of my Traveller campaign, which I wanted to be as detailed and as consistent with known science as possible. Anderson was the master of hard sci-fi and that's why he's my favorite SF author.
EDIT: I never read others' reviews until after I've written my own. Now that I have, it appears that there are actually five volumes in this collection, not three, so it looks like I have some shopping to do. I also don't understand why so many reviewers say this (and volume 1) are "haphazard collections" of stories when the themes seem so clear to me.