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」.
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.
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'