According to Imam al-Qurtuby in his tafsir there are several narrations which are considered as the story behind these verses:
- 'A'isha (Allah be pleased with her) narrated that Allah's Apostle (ﷺ) used to spend time with Zainab daughter of Jahsh and drank honey at her house. She ('A'isha further) said:
I and Hafsa agreed that one whom Allah's Apostle (ﷺ) would visit first should say: I notice that you have an odour of the Maghafir (gum of mimosa). He (the Holy Prophet) visited one of them and she said to him like this, whereupon he said: I have taken honey in the house of Zainab bint Jabsh and I will never do it again. It was at this (that the following verse was revealed): 'Why do you hold to be forbidden what Allah has made lawful for you... (up to). If you both ('A'isha and Hafsa) turn to Allah" up to:" And when the Prophet confided an information to one of his wives" (lxvi. 3). This refers to his saying: But I have taken honey. (This is the hadith version of sahih Muslim)
- An other version where Hafsa (May Allah be pleased with her) was the woman whom gave her husband the honey. And the "conspiracy" was between 'Aisha and Sawdah (May Allah be pleased with both of them) (here a link to the version of sahih Muslim). Note in a narration of ibn Abi Malykah on the authority of ibn 'Abbas (May Allah be pleased with both of them) it was Sawdah whom gave him the honey, others narrated that it was Um Salamah.
- A different story, where the Messenger of Allah (peace be upon him) has declared a slave (some say it's Maria al-Qibtiyah -the Coptic-) as prohibited to him. (Here a reference from sunan an-Nasa-i)
Al Qurtoby said the most sane or strongest of these narration is the first and the weakest the second and he quoted some narrations of Maliki scholars which seem to support the 3rd narration to some extent. (Here's a link to an English translation -apparently summarized- of tafsir ibn Kathir, where you may find the same narrations maybe with more details, as especially the 3rd narration quoted in tafsir al-Qurtoby is much longer than the version of sunan an-Nasa-i).
According to that either the Messenger of Allah (peace be upon him) declared honey as a prohibited food to him, or he has declared his slave girl as prohibited to him. Note that according the hadith he made an oath on that, and he asked the wife he was addressing when he made this oath to keep it a secrete, but she didn't keep the secrete and that is what is referred to in the verses (65:1-5).
The major meaning of the verse -you are asking about- is that one shouldn't prohibit to himself what is made lawful (halal) for him, just to get the approval of others -no matter who they are-. While the next verses are explaining what is to do if one made an oath and prohibited to himself something which is lawful to him for the same purpose (approval of others).
To answer the rest of your questions briefly:
And cant we promise something to ourselves like I will not eat meat?
Yes we can, but it shouldn't be an oath as it apparently was in the case of the story behind (66:1-2).
Can't we promise something to our wives to protect our marriage?
This isn't really covered by this verse, but of course it should be permissible as Allah says in (4:19):
... And live with them in kindness. ...
And our Messenger used to say:
"Indeed among the believers with the most complete faith is the one who is the best in conduct, and the most kind to his family." (Jami' at-Tirmidhi)
"The best of you is the one who is best to his wife, and I am the best of you to my wives." (Sunan ibn Majah)
One could say the story of the verse shows that our Messenger practiced this, as he wanted to treat his wives with some kind of kindness, when declaring that something they dislike is prohibited to him!
I hope this was helpful, feel free to ask, if there's still something missing in this answer!
And Allah knows best.