The book is just what the subtitle says. He describes the people involved, and cites memoirs and declassified documents and puts it all together with a explanations of what the engineering and physics involved looked like.
In one episode, Soviet scientists in Leningrad buried their cyclotron to keep it from being stolen or damaged when the Germans approached (the Soviet war effort would have snaffled up the parts too). While the city was still under siege, a group of scientists had to come back into the city, dig it up, and ship the parts out--to a background noise of incoming artillery. You never knew physics could be so exciting.
Stalin was, of course, very well informed about the Manhattan Project thanks to spies, and to a somewhat open environment behind-the-wire in which American scientists were encouraged to share what they were doing with each other. And he really wanted the bomb. Beria was tasked with getting one built.
As plenty of others have noted, the biggest secret was that the bomb could work at all. After reading the book, I have a much better appreciation for how hard it was, and how many details the devil hid in. And why, even after many years, our military still wanted to do nuclear tests. (It isn't to show off.)
Prof Pondrom uses a number of back-of-the-envelope calculations along the way, but you won't hit any serious math until the appendices, and even there it is just linear differential equations. That matches what the bomb-makers had--you need computers to solve the big hairy hydrodynamic equations, and nobody had them.
Some things are still classified, but there's a lot of detail here.
From Appendix F: "This presentation has been at the level of an undergraduate physics course. If the reader works through this Appendix, and follows the arguments, it does not mean that he/she can build a nuclear weapon in the basement. Weapons design drawings require much more careful analysis than is presented here."
In case you were worried...
Yes, read it.