I just realized, could "run" in this instance refer to a political campaign as well as actual running? What made me think of this right now was "she" in this sentence being Hillary and "the man" being either Donald Trump or Bernie Sanders, depending on if you're talking about the primaries or the actual election.
Note that this difference is reflected in English too, and probably came into Irish from English - long before overt public campaigning was a factor in elections, candidates for election to parliament "stood for election".
In modern parlance, candidates in Ireland and the UK now "run" and "stand" for election at the same time - how much of this is an influence from coverage of US elections, and how much is just a reflection of the "horse-race" nature of election coverage is anyone's guess, but the NEID suggestions for this type of horse-race related election idioms ("in the running", "making the running") don't use rith:
"she's still in the running for the post" - tá sí san iomaíocht don phost fós, or tá seans aici ar an bpost go fóill
"we're making all the running in this campaign" - is muidne atá ag déanamh na hoibre ar fad san fheachtas seo
When it comes to "running a campaign", rith only turn up in one of five examples:
"how campaigns are run" - an bealach a ritear feachtais
"they ran a high-impact campaign" - bhí an-tionchar ag a bhfeachtas ar an bpobal or chuaigh a bhfeachtas i gcion go mór ar dhaoine
"the company ran a mixed media campaign" - chur an comhlacht feachtas meán measctha i gcrích
"he ran a successful campaign" - bhí feachtas rathúil ar bun aige
The OED notes that the “compete as a political candidate” meaning of “run” is “originally and chiefly U.S.” — its first appearance is in a Virginian newspaper from 1826 — though that “and chiefly” was stated before the introduction of the Web. It was listed as an evolution of the “compete in a race” meaning, such as in describing horses.
The “running a campaign” uses seem to be closer to the “operate” meaning of “run”.
?? I'm not sure what "elope" has to do with running.
The Irish for "elope" is éalaigh, which is generally translated as "escape".
For "run away", you can use éalaigh or imigh. here are some examples from in the NEID:
"she ran away with him" - d'éalaigh sí leis, d'imigh sí leis
"he ran away from home" - d'éalaigh sé ón mbaile
"she'd run away and leave it to others" - d'imeodh sí léi agus d'fhágfadh sí ag daoine eile é
Note the preposition in that last example - d'imeodh sí léi - "she'd run away with her(self)"
And just as "jogging" is not the same as "running", rith doesn't means "jog" - the irish for "jog" is bogshodar (literally "soft-trot") though you're probably at least as likely to hear ag jogáil.
The answer you have been given is the answer: because that's how you say it in Irish. No amount of translating each word into English is going to produce an English direct equivalent or explain why it's THAT preposition that is used. Imagine being asked to explain to a French speaker why the expression "I'm down with it" doesn't have anything to do with vertical distance. It just IS and if you want to learn a language there is a certain amount of just learning how certain concepts are expressed in that language. There IS no reason, it's just how you say it.
Ok, first, in the intervening time since I posted this, I've been continuing my exploration of the phrase, and found that the root word is "faoi" which means "about." The literal translation seems to be "about her," or some such, both of which is information I didn't have, so your answer is factually useless.
As they say, "there is no such thing as a "dumb question" but there's such a thing as a dumb answer.
Now, I've had actual Irish teachers explain to me quite a number of Irish idioms that don't translate into english, take for example "Tá ceapaire agam " which literally means "a sandwich is at me." It's a little stilted, but makes sense if you think of "at" as the location. Where is the sandwich? It's at me, because I have it. But agam doesn't translate into English, right? It doesn't mean "I have," but providing this information makes it possible to start forming an understanding of the idiom, and as is especially important, other forms: you have, she has, he has, which all follow the same structure. Of course, agam itself is a contraction, so that part needs explanation as well. There's always worthwhile information that you could impart, that helps solidify understanding of the idiom, which is what education is about. It's never helpful to say "it is because it is."
Now, I'm still looking at this sentence, and after some internet research, it seems there's an english idiom that's pretty close: You could say "She's all about running with that guy," meaning that running with that guy is her main interest at this point. "What does she intend to do?" "Well, she's all about running with that man" So, knowing that gives me a mnemonic framework with which to remember the idiom, and it also gives me enough information so that if someone were to use fúithi in another context, it might actually occur to me that it has something to do with being "about her" or "she is about" instead of "she intends" and I might actually figure it out through context.
See? actual answers. Except that I didn't get them here, I had to waste time fighting with people who are deliberately unhelpful. and then go out into the internet and try to find the information I was looking for. If that's what it's like in the discussion section, I might as well cancel my subscription to Duolingo.
My apologies for crediting you with the intelligence to ask the question that you actually wanted the answer to, and for assuming that, by the time you had been doing Duolingo on a daily basis for 16 months, and had completed the whole tree to level 5, you didn't need a basic, level 1 explanation of how Irish uses prepositions with bí to express various concepts that use a verb in English.
You specifically asked "How does fúithi turn out to mean "she intends"".
It turns out that that isn't what you wanted at all - you wanted a makey-up mnemonic that makes sense to an English speaker but that has absolutely nothing to do with why an Irish speaker uses fúithi to say "she intends".
I didn't tell you that tá fúithi rith means "she's about to run", or "she's about running", because that would be a lie. It's also a lie that Tá ceapaire agam means "a sandwich is at me", as far as Irish speakers are concerned, but it's a comfortable crutch for adult English speakers who are absolute beginners at Irish (I was actually flabbergasted when I first encountered it on Duolingo).
"Apologies"? "Credit you"? Boy, you just never stop. Yes, I specifically asked "How does fúithi turn out to mean "she intends." It's a simple question, and right from the get go you have been snarky about it. I guess I assumed that being in a second language learning discussion, I wasn't jumping into the comments section of a political discussion, and there might actually be some helpful people here. Now, the person who taught me that that Tá ceapaire agam means "a sandwich is at me" is a native Irish speaker, who also teaches Irish for a living. If you look around the internet, you can actually find that exact description. And they pointed out that a native Irish speaker doesn't bother to think of what the phrase actually means they just use it, but if you actually stop and think about it you can see that not only is that exactly what it means, but that it makes sense, even in english. It's not a "comfortable crutch," it's actual understanding of the structure of the language, which is important if you actually plan on understanding the language, instead of just memorizing phrases by rote.
If you think about my question: "How does fúithi turn out to mean "she intends?" You'll see that I didn't just ask: "what does fúithi mean?" It's subtle, so maybe you missed it, but the phrase "how does it turn out to mean?" has a nuanced meaning that implies that I already know it's an idiom. It doesn't just translate, it "turns out to mean" something different than the translation. But I didn't ask a snarky question, I asked a simple question, and immediately got a snarky, useless answer. And my response yesterday is not a "makey-up mnemonic," it's an actual mnemonic framework based on the meaning of the words as they are actually translated, using an etymological framework for a similar idiom. In all likelihood, that's how this idiom came into being in the first place. In other words if you think about it, it makes sense in the same way that has been previously explained by a native Irish speaker who teaches Irish for a living. But at this point, I'm done with you. I don't expect you to make any attempt to read, understand, or make use of anything that I've said. I expect you to read to find things to continue to make self indulgent, arrogant and self-superior remarks to prop up your own ego. Gee, maybe duolingo has a way to block a poster.
How does gorm turn out to mean "blue", how does "kicked the bucket" turn out to mean "died"?
tá fúithí turns out to mean "she intends" because that's one of the ways that Irish speakers say "she intends", just as tá ... aici means "she has ...", and tá uirthí means "she has to".