Your understanding of it being a child's phrase is correct. It is used by PolitiFact in a deliberately childish and mocking way.
You wouldn't use it yourself in conversation, unless you were also trying to be deliberately childish.
In English, we use several metaphors to express this feeling which arises from being forbidden to speak.
- pent-up anger, trapped like an animal in a pen
- bottled up anger, trapped like a fluid in a sealed container
- stifled or smothered anger, stopped from healthy flow like a person who is suffocating
Bridle? Maybe. Literally, bridle, n., is the device for controlling a horse's head. Bridle, v., can mean to control a person or to control one's own visible reactions, as a rider controls a horse using a bridle (trans.). Or, it can mean to react with anger (not necessarily hidden) when one's will is interfered with (intrans.). The problem, then, is that bridled anger may be misunderstood to mean the act of bottling up an emotion, when what you meant was the resulting feeling.
Responding to the follow-up question about where we locate stifled anger: emotional reactions are located in the stomach not because of a cultural metaphor but because of a physical cause. Release of the stress hormone cortisol stimulates the body's “fight or flight” systems and disrupts other systems which might compete for resources, notably digestion. Extreme ongoing emotion can even cause evacuation of the digestive tract and bladder (the person may vomit, urinate, and defecate uncontrollably). This is thought to be an adaptation which enhances survival when one's life is threatened by an attacker.
In English, we often describe the experience of cortisol's effects as a visceral feeling, butterflies in the stomach, knot in the stomach, or sinking feeling in the pit of the stomach.
Several emotions can trigger the stress reaction, such as fear, anger, worry, and excitement. In the case of pent-up anger, the stress is continuous because the situation is unresolved, so we are likely to feel a knot in the stomach.
Best Answer
From The last will and testament of Gen. George Washington (1800)...
Note that words like intention, meaning, reason, point all overlap in various contexts. And as this 1727 instance shows, point has been around with this reason, argument sense for a long time.
Presumably what OP finds "odd" is the word being, in a construction where the more obvious verb form would simply be is. To be fair, Robert Noggle at Central Michigan University also says "the reason being [some reason]" is somehow invalid (he certainly doesn't like it), and maybe there are others who feel the same. But I disagree - it seems like perfectly standard English to me.
Arguably in some contexts the present continuous "being" emphasises the ongoing nature of whatever is being referenced more than simple present "is". For example, "My point being that X" could imply some "insistent" sense of "My point continues to be / has always been that X".