Why does a local inertia frame Lorentz transform when going from $x$ to $x + dx$?

This is because a manifold doesn't have vectors in and of itself. You have to supplement an abstraction called the tangent space. That is a vector space. Curves arise as the most natural objects to define on a manifold. And when you move from point $x$ to $x+dx$ along a curve, and if you're using the tangent space all along this displacement to find your bearings on the landscape, you must define a way to say when vectors are moving parallelly or not. You can picture this like trying to move an arrow that describes your motion when going from a point to another on the surface of an apple. The arrow will have to rotate, to adapt to the new "tangent space". In case your manifold is locally Minkowskian, it will have to Lorentz-rotate. Tony Zee is notorious for a pedagogical style in which he sacrifices some level of mathematical rigour in order to provide an intuitive picture, and he's a master at this. The really rigorous concept that takes you from vectors at $x$ and those at $x+dx$ is the connection.