Other people have covered explaining the actual meaning of the theory this paragraph referenced. I will instead address how the quoted paragraph was intended to be interpreted, such that it does in fact correctly describe the theory.
During the first half of the twentieth century, mathematicians such as <PERSON>, <PERSON>, and <PERSON> discovered that certain basic problems ¨ cannot be solved by computers. One example of this phenomenon is the problem of determining whether a mathematical statement is true or false.
The crucial point for this segment is that "a mathematical statement" is not referring to any particular specific statement, nor even to a not-yet-specified statement. "A mathematical statement" is describing an input to the problem, that may be given any arbitrary value that matches the description, and truly solving the problem requires specifying a solution that will work for every conceivable possible value of that input.
This task is the bread and butter of mathematicians. 682
It seems like a natural for solution by computer because it lies strictly within the realm of mathematics. But no computer algorithm can perform this task.
This segment confusingly uses the phrase "this task" to mean different things in different places, only one of which actually matches the problem described immediately before it.
The task of determining whether an individual specific mathematical statement is true or false is indeed the bread and butter of mathematicians, and that application of the problem to individual input values is what the first use of "this task" refers to.
The other use of "this task" refers to the general case I described above, of solving the problem for every conceivable input value, so that your solution is guaranteed to produce a correct answer no matter what input value you apply it to.
The general case task is impossible for computers, and it is also impossible for mathematicians. In fact, it's simply impossible, period. That result, the fact that the general case cannot be completely solved by any means, is the really remarkable thing about this.
Among the consequences of this profound result was the development of ideas concerning theoretical models of computers that eventually would help lead to the construction of actual computers
This section is going off on another topic, irrelevant to understanding the issue asked about here. 716