"They hear that she speaks Danish."
Translation:De hører at hun snakker dansk.
The exact same difference as between speak and talk. The exact differences between the understanding of the words properly differ between people but in my understanding 'speak' and 'snakke' implies less formality and less meaning behind the words than 'talk' and 'tale'. But for translation it is very easy just use 'snakke' for 'speak' and 'tale' for 'talk'.
Though, if you're going on the formal/informal distinction it's the other way round in English: "speak" is the more formal word, "talk" the less formal. So, tale : speak and snakke : talk.
Same remark as in case of jeg ser at han spiser sin mad.. the at/that seems redundant to me.
Why is it not "De hører, at hun snakker/taler dansk"? Especially since the question setter has gone to the trouble of writing "They hear that she speaks Danish", when "They hear she speaks Danish" would have done. N.B. I ask because in the exercise I am currently doing the "correct answer" supplied is "De hører hun snakker dansk" (with no "at").
I believe that "De hører hun snakker dansk" is incorrect Danish. It appears to be corrected now, to "De hører, at hun snakker/taler dansk", which didn't work for you earlier. I think Danish also requires the comma in the position where you put it, but I am not sure.
my dude you are on a 566 day streak how is this possible ive been on for two days
Short useless answer : Day by Day.
John Arnold, that is in top 3 longest Duolingo streaks ever, did it by starting each day with it. https://twitter.com/duolingo/status/1017433362460528640
Short useful answer : you can try ideas from https://markmanson.net/goals