The ones I have seen do this are more of a "beg" than a "threat".
They probably don't have the best understanding of human interaction and their wording is frequently not the best. Usually there is a sense of panic for them as they probably really want to keep the site going but have found out that financially a site costs more than they originally thought.
Some of those also are ones that started a site thinking it was an instant pathway to riches, not realizing that those days are long gone and they have figured out that you have to actually now address running a site that you want to make income on as a business, with a business plan, solid funding resource and adequate staffing. And when the reality sets in that you have to spend money to make money panic starts as they had assumed it would be free/easy money income stream.
If you are running your site as a hobby, then you still need to treat it as a business, but the funding is assumed to be present since hobbies usually cost and are budgeted for. But by the same token, those hobby sites are subject to not being able to be paid for if life circumstances change. Some then get to the point of desperation because running a site has become part of their life and if they lose that, they feel that they have lost a major part of their life so they start trying to get others to help fund it.