Temporal Knowledge Graph (TKG) forecasting faces significant challenges due to distribution shifts and poor inductive generalization in parametric models. While Large Language Models (LLMs) offer potent semantic reasoning, existing LLM-based approaches struggle with implicit modality alignment and suboptimal graph linearization, failing to capture complex topologies without retraining. To bridge this gap, we propose ExE-LLM, a training-free, test-time adaptive framework that reframes TKG prediction as explicit evidence-driven reasoning. Our core philosophy is to decouple topological calculation from semantic reasoning: a heuristic module translates latent graph signals into natural language evidence, enabling the LLM to perform multi-source judgment. ExE-LLM incorporates a task-aware scheduler for test-time adaptation, a heuristic synthesizer for explicit modality alignment, and a self-diagnosis module for iterative optimization. Extensive experiments on four benchmarks demonstrate that ExE-LLM achieves SOTA performance in inductive settings, significantly outperforming fully trained graph neural networks without updating LLM parameters. The source code is available at https://github.com/JENLISA4EVER/ExE-LLM.