I can't understand this. In symphony you hear the word "movement" to refer to independent sections. But it seems that it's not really that relevant. Movement almost everywhere has something to do with motion. Is this nomenclature a kind of metaphor for something in music, like moving from one section to another?
The earliest meaning of movement in music was how it moved: it's tempo or relative speed (probably referring to the natural movement associated with the beat of the tempo). For instance, the OED quotes an entry from Baily's 1721 dictionary:
Similarly, Henry Purcell's Sonnatas of III Parts (1683) in its preface speaks of "Allegro, and vivace, a very brisk, swift, or fast movement."
As musical forms with multiple sections, such as symphonies, evolved, each of the sections generally had a different tempo, or movement (which was a main reason for having distinct sections). One would refer to a section of a work by its tempo, e.g. the allegro movement or the largo movement. A natural evolution was then to speak of the several movements of the work, that is, the several sections of different tempos.
The usage of movement in reference to tempo per se has largely died out, but the reference to movements as sections of different tempo has continued. We still often find a tempo term (e.g. vivace) as the title or part of the title for a movement.
By definition movement indicates the transfer of an object from one place to another. In music that object could be considered an emotional or conscious state of being, therefore the music may move you.
The musician(s) go through the act of moving physically through the music, making the music the vehicle to that change.
Of course, that is all very esoteric. The fact is, "movement" is just what it's called. "A movement is a self-contained part of a musical composition or musical form." (Wikipedia)
A Sonata, which itself may make up a single movement within a greater composition, it is itself made up of 3 main sections (Development, Exposition, Recapitulation) which can be considered movements themselves within the Sonata form.
Wikipedia is a pretty good start if you want to learn more.