Translation:She accepts that these diseases exist.
The definite article "die" (or "der" or "das") can mean "the", "this", "that", "these" or "those" depending on which context fits the best. For this sentence, both "these" and "the" are accepted by DL.
Right! And just to add to your remark, the meaning in spoken German often depends on the way the words of the sentence are stressed. Unfortunately, stressing the proper words is something that the Duolingo language model doesn't do very well right now, so it's hard to get a good sense of this language aspect here. But if, for example, "die" in this sentence is stressed, it would translate more accurately to "these" or "those" than "the".
Both bestehen and sein can mean "to exist", although each of those words also has other meanings that the other does not.