This is a question that many people talk about. And I think it is important because incorrect responses can mislead individuals and significantly influence his/her attitude to studying mathematics.

Let me quote the following two to express my opinion.

The answer is an emphatic NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”. And yes, a reasonable amount of intelligence, patience, and maturity is also required. But one does not need some sort of magic “genius gene” that spontaneously generates ex nihilo deep insights, unexpected solutions to problems, or other supernatural abilities. ---Terry Tao

You ask me if an ordinary person—by studying hard—would get to be able to imagine these things like I imagine. Of course. I was an ordinary person who studied hard. There's no miracle people. It just happens they got interested in this thing, and they learned all this stuff. They're just people. There's no talent or special miracle ability to understand quantum mechanics or a miracle ability to imagine electromagnetic fields that comes without practice and reading and learning and study. So if you take an ordinary person who's willing to devote a great deal of time and study and work and thinking and mathematics, then he's become a scientist. --- Richard P. Feynman

Both scientists agree that one does not "magical" or "miracle" talent to be a scientist. Maybe you will argue that these two persons (Tao and Feynman) are already geniuses, they just don't understand how hard for an ordinary person to follow their thoughts. If you have such an opinion, I have to say you are wrong. If you understand in detail how they think and teach, you will find that they surprisingly know ordinary people too well! (I should put some examples here.)

And I can understand that, there are plenty scientists who are very good at researching or publishing papers, but their lectures are so messed up that it’s hard for normal students to understand. (I have lots of such experiences as a student, to be honest.) This may create the illusion that only geniuses can understand mathematics. But in fact, these hard-to-understand people are not doing well in some sense of mathematics. Here I refer readers to the article What is good mathematics, which discusses many aspects of mathematical ability.

You may still argue that I just collected something that support my opinion and ignored others's objections. You're right, there are also many domestic mathematicians who warn that if you don’t have a high IQ, don’t study mathematics, or don’t choose mathematics as your major. Believe it or not, these kind of researchers say like that probably because they don’t know how to teach an absolute beginer, they think their explanation is very well which is far from the truth. I have lots of words to speak out here, but I'd better keep polite. After all, I'm just an ordinary student who has no paper in PhD period yet (if you really care).

However, what I believe is: Every thing in mathematics can be understood, if only I pay enough time into it like Feynman said. Learning mathematics and understanding mathematics is itself a wonderful experience, which is irrelevant to anything else.

Reference

• Does One Have to be a Genius to Do Maths?
• What is good mathematics?
• Feynman-"I was an ordinary person who studied hard"