Matter and antimatter particles are always produced as a pair. This particle physics' process is called pair production. If antimatter is detectable, it means that the
antiparticle of a matter-antimatter-particle-pair was able to survive long enough to be detected, for instance due to relativistic effects, as explained e.g. by the Hyperphysics page on muons:
The measurement of the flux of muons at the Earth's surface produced an early dilemma because many more are detected than would be expected, based on their short half-life of 1.56 microseconds. This is a good example of the application of relativistic time dilation to explain the increased particle range for high-speed particles.
You will now correctly say that a muon is not antimatter, but I just used the muon as example that some particles (or antiparticles) may be detected even if they have a short half-life.
So, what are the sources of antimatter in the magnetosphere? Strictly speaking, the source (in the sense of "location") of pair creation will probably be in the magnetosphere itself, as said by atmosphericprisonescape:
Anywhere where highly energetic reactions take place whose center-of-mass energy surpasses the rest-mass energy of 2*times some particle, this particle and its anti-particle counterpart can be created.
I suppose the real question is: How is their enough energy for frequent pair production in the magnetosphere?