It's totally rigorous and starts from, "the ability to read English and to think logically -- no high-school mathematics, and certainly no advanced mathematics."
Mathematics is foremost a conceptual subject rather than a mechanical one, and it is immaterial that the reader have firsthand experience that all the theorems are proven. As one learns mathematics, it soon becomes apparent that there will always be gaps in her or his knowledge, and that is therefore best to skip steps that s/he believes could be done in principle.
I personally believe that Landau was caught up in the spirit of the times and optimistically believed that math could be built up from "first principles". The famous kickstart to this is Hilbert's 1900 presentation. And it certainly continued up through Nicolas Bourbaki.
In fact, Landau's mathematics is presented in a somewhat archaic style and his proofs are extremely hard to follow in spots, as if he is making unstated assumptions. Overall, it is an interesting, but ultimately thankless, task to go through that book. It is a mostly a historical curiosity. The same can be said of Hardy's "Course of Pure Mathematics", which was recommended elsewhere in this thread. I find it hard to believe that anyone who recommends these books have actually read them.
To the OP, while I can relate to the goal from personal experience, after decades of going down a similar path, I can tell you that the history of math is very messy. Our textbooks and notation reflect this messiness. My recommendation is to dive into whatever part strikes your fancy, although it may help to start from where you are. For instance, if you program, you might want to get a book on physics in game programming or learn Haskell.