Client perceptions of the therapeutic alliance are critical for counseling effectiveness. Accurately capturing these perceptions remains challenging, as traditional post-session questionnaires are burdensome and often delayed, while existing computational approaches produce coarse scores, lack interpretable rationales, and fail to model holistic session context. We present CARE, an LLM-based framework to automatically predict multi-dimensional alliance scores and generate interpretable rationales from counseling transcripts. Built on the CounselingWAI dataset and enriched with 9,516 expert-curated rationales, CARE is fine-tuned using rationale-augmented supervision with the LLaMA-3.1-8B-Instruct backbone. Experiments show that CARE outperforms leading LLMs and substantially reduces the gap between counselor evaluations and client-perceived alliance, achieving over 70% higher Pearson correlation with client ratings. Rationale-augmented supervision further improves predictive accuracy. CARE also produces high-quality, contextually grounded rationales, validated by both automatic and human evaluations. Applied to real-world Chinese online counseling sessions, CARE uncovers common alliance-building challenges, illustrates how interaction patterns shape alliance development, and provides actionable insights, demonstrating its potential as an AI-assisted tool for supporting mental health care.