Recommendation systems have been essential for both user experience and platform efficiency by alleviating information overload and supporting decision-making. Traditional methods, i.e., content-based filtering, collaborative filtering, and deep learning, have achieved impressive results in recommendation systems. However, the cold-start and sparse-data scenarios are still challenging to deal with. Existing solutions either generate pseudo-interaction sequence, which often introduces redundant or noisy signals, or rely heavily on semantic similarity, overlooking dynamic shifts in user motivation. To address these limitations, this paper proposes a novel recommendation framework, termed M-$LLM^3$REC, which leverages large language models for deep motivational signal extraction from limited user interactions. M-$LLM^3$REC comprises three integrated modules: the Motivation-Oriented Profile Extractor (MOPE), Motivation-Oriented Trait Encoder (MOTE), and Motivational Alignment Recommender (MAR). By emphasizing motivation-driven semantic modeling, M-$LLM^3$REC demonstrates robust, personalized, and generalizable recommendations, particularly boosting performance in cold-start situations in comparison with the state-of-the-art frameworks.