To promise the moon: to say that you will do more than is possible for you to do. To promise more than can be delivered.
#1 - A politician giving speeches to potential voters
In his speech, the politician promised the moon to the people so that they would vote for him. He promised that he would reduce their taxes and that under his leadership crime would drop.
#2 – A boyfriend asking his girlfriend to marry him
The man promised the moon to his girlfriend. He told her that he would become wealthy and that in two years she would be able to quit her job and focus on her art. He told her that they would go on exotic vacations and have a winter home in Spain.
To promise the moon: This idiom is used when something is unlikely to happen even though the speaker has said that it would. The speaker is presenting an attractive future in the hopes that he will also appear attractive.
In example #1, the politician is trying to convince people to vote for him by promising attractive things in the future even though he may have no intention of even trying to fulfill that promise.
In example #2, the boyfriend is hoping his girlfriend will accept his proposal for marriage. Perhaps he fears that she will reject him, so he is making her future with him look very attractive. He may feel especially positive about his future. It could be that he is not trying to deceive her, only afraid that he may lose her. He wants her to feel that he will work hard for her and make their lives together as wonderful as he can. He may be unrealistic about what he can do for her.