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:
- Smartphones have their roots in music players, aiming for the chic, slick design as well as simplicity and portability over massive computing power or general utility.
- Stack Exchange has its roots in the generic forums, blogs, and boards that can be found all around the internet, but was carefully pruned to serve a Q&A role.
- Modern gaming has its roots in arcades, a service providing visual, interactive entertainment for money, often in a group context.
- Most of the Science Fiction and Fantasy Genres have their roots in the early to mid 20th Century, when it was becoming radically cheaper, easier, and faster to write large amounts of material and communicate them over long distances.