What is the meaning of a woman giving herself in marriage to a man, and was the Prophet's (peace be upon him) rejection of the woman who offered herself to him due to his not needing her, or for another reason?

A woman gifting herself to a man means her marriage to him without a dowry. This was a privilege exclusive to the Prophet, peace and blessings be upon him, not his nation, as stated in the Almighty's words: "And [also any] believing woman who gives herself to the Prophet if the Prophet wishes to marry her — [this is] exclusively for you, excluding the [other] believers." (Al-Ahzab/50).

It has been narrated that a woman offered herself to the Prophet, peace and blessings be upon him. He looked at her, then lowered his head and did not decide anything regarding her. A man from the Companions then stood up and asked to marry her if the Prophet had no need for her. The Prophet, peace and blessings be upon him, asked him about a dowry, and since he found nothing, he married them by virtue of what the man possessed of the Quran.

The Prophet's look at the woman indicates the permissibility of looking at a prospective fiancée. As for his turning away from her, it relates to acceptance and peace of mind, not necessarily to her beauty or his desire.