What is meant by a woman offering herself to a man, and is that permissible?

Firstly:

What is meant by a woman offering herself to a man is that he may marry her, but without a mahr. This was only for the Prophet (blessings and peace of Allah be upon him) and not for any of his ummah.

Allah, may He be exalted, says (interpretation of the meaning):

{O Prophet, indeed We have made lawful to you your wives to whom you have given their due compensation … and a believing woman if she gives herself to the Prophet [and] if the Prophet wishes to marry her, [this is] only for you, excluding the [other] believers} [al-Ahzaab 33:50].

Al-Baji (may Allah have mercy on him) said: There is no difference of scholarly opinion regarding the fact that it is not permissible to get married without giving a mahr for anyone other than the Prophet (blessings and peace of Allah be upon him). The basic principle concerning that is the verse in which Allah, may He be exalted, says:

{O Prophet, indeed We have made lawful to you your wives to whom you have given their due compensation … and a believing woman if she gives herself to the Prophet [and] if the Prophet wishes to marry her, [this is] only for you, excluding the [other] believers} [al-Ahzaab 33:50].

Here Allah, may He be exalted, stated that that was only for the Prophet (blessings and peace of Allah be upon him), to the exclusion of all other believers, so that is not permissible for anyone except him.

Based on the meaning of this hadith, the woman said to him: O Messenger of Allah, I offer myself to you. He did not criticise her for that, and if it was wrong, he would have objected to it and would not have approved of it, because the Prophet (blessings and peace of Allah be upon him) could not approve of any falsehood. Moreover, when someone else stood up and asked to marry her, he did not let him marry her without a dowry, even though he was poor and in need, and had nothing to give her as a dowry, to the extent that the Prophet (blessings and peace of Allah be upon him) married her to him for what he had memorised of the Qur’an. If it were permissible for anyone other than the Prophet (blessings and peace of Allah be upon him) to marry without giving a mahr, he would not have prevented it in this case, even though the man is very poor and in great need."(Al-Muntaqa Sharh al-Muwatta’  3/275).

Secondly:

Al-Bukhari (5030) and Muslim (1425) narrated from Sahl ibn Sa‘d that a woman came to the Messenger of Allah (blessings and peace of Allah be upon him) and said: O Messenger of Allah, I have come to give myself to you (in marriage). The Messenger of Allah (blessings and peace of Allah be upon him) looked her up and down, then he lowered his head. When the woman saw that he had not made any decision about her, she sat down. A man among his companions stood up and said: O Messenger of Allah, if you have no need of her then marry her to me. He said: “Do you have anything?” He said: No, by Allah, O Messenger of Allah. He said: “Go to your family and see if you can find something.” So he went, then he came back and said: No, by Allah, O Messenger of Allah, I did not find anything. He said: “Go (and see if you can find something), even if it is a ring of iron. So he went, then he came back and said: No, by Allah, O Messenger of Allah, not even a ring of iron, only this izar (lower garment) of mine – Sahl said: he did not have a rida’ (upper garment) – and she may have half of it. The Messenger of Allah (blessings and peace of Allah be upon him) said: “What will she do with your izar? If you wear it she will not have anything of it and if she wears it you will not have anything of it.” The man sat down, and after he had sat for a long time, he got up (to leave). The Messenger of Allah (blessings and peace of Allah be upon him) saw him turning away, and he ordered that he be called to him. When he came, he said: “What do you know of the Qur’an?” He said: I know Surah such and such, and Surah such and such, and Surah such and such – and he listed them. He said: “Do you recite them by heart?” He said: Yes. He said: “Go. I give her to you (in marriage) for what you know of the Qur’an.”

The Prophet (blessings and peace of Allah be upon him) only looked at her because his position was like that of a suitor, who is allowed to look at the woman to whom he wants to propose marriage, then if he likes her he may marry her, and if he does not like her, he must avert his gaze.

Ibn al-Jawzi (may Allah have mercy on him) said: “Then he looked her up and down” means that he looked at her face, then lowered his gaze to look at what is below that. This indicates that it is permissible to look at the woman whom one wants to marry. He only did that because it was possible that he might have wanted to marry her, then when he decided not to marry her, he lowered his head."(Kashf al-Mushkil 2/270).

An-Nawawi said: This indicates that it is permissible for the one who wants to marry a woman to examine her."(Sharh Muslim 9/212).

As for the Prophet (blessings and peace of Allah be upon him) not wanting to marry her, we cannot be certain about the reason for that. How many suitors look at a woman then change their mind, despite her beauty. The matter is not limited to the issue of beauty or desire; rather what matters is whether the suitor likes her and feels at ease with her. The Prophet (blessings and peace of Allah be upon him) probably did not like this woman, or he did not feel any inclination to marry her, or there may have been other reasons.

And Allah knows best.