Multi-branch deep neural networks like AASIST3 achieve state-of-the-art comparable performance in audio anti-spoofing, yet their internal decision dynamics remain opaque compared to traditional input-level saliency methods. While existing interpretability efforts largely focus on visualizing input artifacts, the way individual architectural branches cooperate or compete under different spoofing attacks is not well characterized. This paper develops a framework for interpreting AASIST3 at the component level. Intermediate activations from fourteen branches and global attention modules are modeled with covariance operators whose leading eigenvalues form low-dimensional spectral signatures. These signatures train a CatBoost meta-classifier to generate TreeSHAP-based branch attributions, which we convert into normalized contribution shares and confidence scores (Cb) to quantify the model's operational strategy. By analyzing 13 spoofing attacks from the ASVspoof 2019 benchmark, we identify four operational archetypes-ranging from Effective Specialization (e.g., A09, Equal Error Rate (EER) 0.04%, C=1.56) to Ineffective Consensus (e.g., A08, EER 3.14%, C=0.33). Crucially, our analysis exposes a Flawed Specialization mode where the model places high confidence in an incorrect branch, leading to severe performance degradation for attacks A17 and A18 (EER 14.26% and 28.63%, respectively). These quantitative findings link internal architectural strategy directly to empirical reliability, highlighting specific structural dependencies that standard performance metrics overlook.