Translation:How have you been?
Could we have some additional grammar notes at this point. The comment below breaking down this phrase was useful. Who would know that -me- indicates the present perfect tense, or that shinda means "win" without some notes. Duo lingo is not some tourist phrase book for people to memorize and parrot. The virtue in the system is that one really learns actual grammar.
I agree with your message in spirit, but I think there is something to be said for taking things a bit slow. These points of grammar will be explained in later skills, I'm sure, and as the course pointed out, the greetings and similar phrases were put at the start of the course because it's apparently a culturally important thing in Tanzania and perhaps other places where Swahili is spoken.
I also always look for morphological breakdowns of such things right away in the comments, but I can also understand the desire not to make the tips and notes too overwhelming for a new learner.
The basic meaning isn't directly to beat or conqure, but more to overcome or endure. When you use it in this case, what you are overcoming is time, or the day. When you talk about beating someone, what you are overcoming is another person or persons, as with a sports team.
I've also seen the word being used for 'to spend' when talking about time, as in "Tumeshinda siku hizi mjini" - "We spent these days in town". I'm less certain about this last use of the word though.
"Habari ya leo?" is asking specifially about (the news of) THAT day. ("What news of today?") In contrast "Umeshindaje?" can be used even a week later. Let's say I had a difficult exam on Tuesday and Friday or a week later I meet a friend who knows I had that exam - correct me if that's wrong. :)
Literally, it means "how have you succeeded?" But functionally it's a greeting for the evening/ late afternoon.
Probably, it's being taught now because, even before learning about tenses, it's necessary to be able to exchange greetings. The important thing, at a beginner stage, is not to be able to parse the grammatical complexities of this phrase (It is quite complex!) The important thing is that when you hear it, you recognize that you are being greeted, and you respond in a contextually meaningful way. And that, when you meet someone in the evening or late afternoon, you can offer the appropriate greeting, "Umeshindaje?"