Hello, Amirhossein Asgharnia, and welcome to the Grammar Exchange!
In the third edition of Practical English Usage, Swan gives those two examples in entry 580, on page 573.
Amirhossein Asgharnia posted:
1) This discovery means that we will spend less on food.
2) This discovery will mean that we spend less on food.
Since the subordinate clause is after that, so the second one may be right; however, the first one seems more natural for me. I wanted to ask, which one is the right answer, and if both, what do they mean?
They are simply example sentences. There is no "answer" to find. Both sentences are grammatically correct. When Swan wishes to say that a sentence is incorrect, he makes it unmistakably clear that he is doing so by using "NOT" before the sentence and crossing it out.
In order to understand what the sentences mean, we would have to know what "this discovery" refers to, and that information is not provided. Swan simply wants learners to understand that when the future tense is used in the main clause of such a sentence, it is not also used in the subordinate clause:
NOT: This discovery will mean that we will spend less on food.