Humans exhibit remarkable compositional reasoning by integrating knowledge from various sources. For example, if someone learns ( B = f(A) ) from one source and ( C = g(B) ) from another, they can deduce ( C=g(B)=g(f(A)) ) even without encountering ( ABC ) together, showcasing the generalization ability of human intelligence. In this paper, we introduce a synthetic learning task, "FTCT" (Fragmented at Training, Chained at Testing), to validate the potential of Transformers in replicating this skill and interpret its inner mechanism. During training, data consist of separated knowledge fragments from an overall causal graph. In testing, Transformers must combine these fragments to infer complete causal traces. Our findings demonstrate that few-shot Chain-of-Thought prompting enables Transformers to perform compositional reasoning on FTCT by revealing correct combinations of fragments, even if such combinations were absent in training data. Furthermore, the emergence of compositional reasoning ability is strongly correlated with model complexity and training-testing data similarity. We propose, both theoretically and empirically, that Transformers learn an underlying generalizable program from training, enabling effective compositional reasoning during testing.