Translation:I will participate in the Olympics.
If it helps you memorize it:
We exit our city into the Olympic city.
We exit the locker room into the stadium.
We exit what we were before, and step into our new Olympic life.
The 「出」/「で」character can still be thought of as "to exit." The thing your are exiting would be marked with「を」, and this sentence does not state it. Without 「を」, you know that the place you're exiting from is not important. The thing that is important is「に」. We've used「に」with other verbs like「行く」and「来る」to mark the destination of where we're moving to. It makes sense that for「出る」, the place your exiting and moving into, the Olympics, would also be marked with「に」.
That particle explanation is very misguiding and i am suprised to see it everywhere and so well ranked.
Actually, this concept is not weird at all in other languages. In Spanish for example, the verb "salir" can also be used as "to participate" / "to be seen in" and "to leave". This sentence would be translated as "quiero salir en los olimpicos". The reason they translated it as "to participate" in english is because the verb "to leave" doesnt convey "to be seen in..." in english. So you can translate it as "to participate" ONLY when that can be seen by others. It doesn't have anything to do with the particle directly as Genki and other comments suggest.
The general idea is that if whatever you are talking about becomes visible in a place, visible to others, you can use 出る. The reason is that when you leave a place you become visible... outside. When you leave your house I can see you from outside. That is why we use the same word to express "to leave" and "To appear" or to be seen、like in the news or at the olympics. Basically, the act of leaving a place can be observed from inside (as English speakers do) where you are disappearing, or from outside (as Japanese, Spanish or Português speakers do), where you are appearing. In time, the concept evolved and now you dont have to leave anyplace, but if something appears somewhere we use the same word. Usually, but not always, it is accompanied by に just because it is marking the place where others are seing you to appear, just in the same way "salir" in Spanish is accompanied by "en" but not always. In 学校に出ます "I will leave to school" for example, that particle explanation doesn't work. Also, it has nothing to do with being a participant in anything. The thing is that if you participate in a public event you become visible. So one of the translations to English is "to participate". This whole idea also applies to the transitive version 出す。That is why besides "to take out" it also means "to publish" and "to make public" and it is still transitive: そのニュースを新聞に出しました。"i published that news in the paper"
Instead of "to participate" I personally would use "to appear" since it is more useful in sentences like "この問題はテストに出ます" or "este problema saldrá en el examen" or "this problem will appear in the test". Or マリアはニュースに出ました。"Maria appeared in the news". It is easier to grasp the concept with "to appear" than other translations.
Other uses that 出る have that you can also see in exactly the same form in Spanish and Portuguese are to appear, to come out, to be discovered, to come forth, to emerge, to graduate... And many others.
So now you know it doesn't have "two meanings depending on particles" it is just one concept that covers a lot. In fact, there are more than 10 translations in my Japanese English dictionary but just "salir" in spanish, and all related to the same idea. So you will have to choose properly when translating to English where that concept doesn't exist.
This one word 出る, salir, is one of the reasons I like languages so much. Because how it opens your mind to new concepts. Two cultures so seperated in distance got to the same conclusion of seeing the act of leaving from outside. Mind-blowing. すごいですね。
The key here is the particle used. Muehlhiasl's comment demonstrates the difference, but the truth is, in both cases, the verb 出る means exactly the same thing – to go/come out. It will be easier to understand this sentence and other Japanese sentences if you develop a general idea of what the particles mean. 〜に often expresses a sense of direction or a movement towards something like a target or destination. That's why we say things like 水を皿 に 入れる (to pour water into a dish). The main exception to this rule is when you say 〜にある or 〜にいる, but even there, に indicates a point/place of focus.
~を indicates the object of a verb, i.e. the thing or person to which the action is done. So, when you say 部屋(heya)を出る, the act of leaving/coming out/going out is applied to the room. Hence, you 'leave the room'.
I think the problem here is that Duolingo makes us memorise translations without understanding the fundamental – even literal – meaning of each word. That's why you feel like there's a flip-flop. Let me propose another translation that is less natural in English, but which might be easier to understand: 'I/he/she/it will appear/come out in the Olympics.' Does the sentence make more sense to you now? Do you still feel the two meanings are contradictory?
I hope this helps, and that you no longer feel you have to memorise the two meanings: all you need to do is reconstruct them from the components – 〜に出る for 'come/go out in(to) 〜を出る for 'come/go out of /leave (a place)' 〜から出る for 'come/go out from/of'
It's similar to "to step/come out", though, isn't it? These also can mean both "appear" and "leave".
You step out in front of the metaphorical line to show yourself and make an appearance (for example at the Olympics).
And then you step out of the competition when you leave it.
I can't think of a situation which would give rise to the notion of 'leaving the Olympics'. You might boycott them, but that would be a different verb. 'deru' is used and I have seen it in the sense of 'to appear in something' as well as 'to participate in something'. We say, when speaking of the Olympics, 'to participate' so that is how you should translate it.
Exiting (and the same is true for entering) is a transition action, you exit some place and at the same place you enter another.
It is just that 出る (exit) uses the point of view of the place you quit. That place you quit is marked with を but you can also tell the place you "exit to", which is marked with に
Here を is not used; because the speaker doesn't feel important to talk about the place he exits from; we can suppose it is his house, his hometown, his country, something like that;
however, he specifies, with に the place he will be after the exiting : the Olympics.
The idea is that he is going out of his everyday life, into (に) the Olympics.
As others have pointed out, there isn't in English an equivalent expression (a verb from the point of view of the "going out" for talking about the destination); but in other languages similar constructions exist (like in Spanis "salir de compras" (lit. "to go out for shopping"), "salir de fiesta" (lit. "to go out to have fun").
It is a dynamic motion from on e place to another; some languages (like Spanish or Japanese) express it with a verb that attaches to the point of view of the starting point (salir, 出る), while other langues express it with a verb that attaches to the point of view of the ending point ("to appear (participate) in").
To understand a different language, in particular one of a very different linguistic family, it is sometimes helpful to take some height, and dont focus on the translation, but on various aspects of the "whole picture" of the idea being communicated.
(There is, for example, a joke in French, that said the English are strange, because they name their metro stations with names of defeats... well, it's the point of view, the Napoleonic/French victories were English defeats and vice-versa. The events (the battles) were the same of course, but often each language/culture uses a specific angle)