In the page 5 of which Mechanics by written L.D.Landau, this book said "If we were to choose an arbitrary frame of reference, space would be in-homogeneous and an-isotropic. This means that, even if a body interacted with no other bodies, its various positions in space and its different orientations would not be mechanically equivalent. The same would in general be true of time, which would likewise be in-homogeneous; that is, different instants would not be equivalent. Such properties of space and time would evidently complicate the description of mechanical phenomena. For example, a free body (i.e. one subject to no external action) could not remain at rest: if its velocity were zero at some instant, it would begin to move in some direction at the next instant." In the example of above sentence, why is this true "the free body could not remain at rest without the external force" in inhomogeneous and anisotropic space?
You are missing what Landau wrote before that part: "If we were to choose an arbitrary frame of reference, space would be inhomogeneous and anisotropic".
If you choose an accelerating frame of reference, fictitous forces will be present and free bodies will start to move even if no force is acting on them.
As Landau writes, you have to choose a frame in which space is homogeneous and isotropic: an inertial frame.
From Wikipedia (https://en.wikipedia.org/wiki/Inertial_frame_of_reference):
All inertial frames are in a state of constant, rectilinear motion with respect to one another; an accelerometer moving with any of them would detect zero acceleration.