The Movement has its roots in combating colonialism.
What does the expression has its roots in mean? Does it indicate a reason or a time? That is, was the Movement started to combat colonialism or was it started in the colonialism era?
It's an analogy to plants, so if
A has its roots in
A grew out of the
B, inheriting the situation and constituents. Just like a tree draws many of its nutrients from the soil below it,
B will shape
A. In your example, the anonymous "Movement" grew out of "combating colonialism", meaning that the Movement drew deeply from the people, ideas, and resources of combating colonialism. The Movement will probably be defined largely in relation to agreeing or disagreeing with particular points of combating colonialism, though predominantly agreeing with them.
Additionally from the analogy of a plant growing it often has the connotation of a sort of natural progression, that a seed took the fertile "ground" of
B and created this new structure,
A, without there having been any real intention for that to happen.
A few hopefully-illustrative examples:
Uses a metaphor of roots to imply an intrinsic quality and origin of whatever happens to be springing roots.