Now I'm just a simple boy from the farm, so can somebody explain why this is important to teaching mathematics?
Sure, I'll give it a try. The "this" that is supposed to be important are the qualities of being empathetic to, and understanding of, and passionate about, the unique challenges facing people who belong to certain racial, ethnic, or other demographic groups. Although the primary role of a mathematics teacher is to impart knowledge about mathematics, there are many ways and approaches to doing this, some a lot more effective than others. Part of the philosophy of diversity is the belief that a teacher (of mathematics or any other subject) who has a good understanding of and empathy towards those groups will be a much more effective teacher to students who are members of those groups, and, for example, in addition to successfully imparting the desired technical knowledge, would also have the ability to inspire their students, teach them additional useful values and habits, and overall improve their educational and life outcomes.
Aren't people, well people?
Yes, that is technically correct but is probably not a very helpful way to look at a question such as what makes someone a good candidate for a teaching position in a community college. The sad fact is that humans are programmed in many subtle ways to treat different people in different ways. The ability and desire to be aware of such tendencies and overcome them are very desirable qualities for a lecturer in a country as heterogeneous as the U.S.
And if a person knows how to do deal and relate well with others, knows how to respect other's differences and is a good teacher, what more is there to say really?
I think this question carries a premise that a person either relates well to all "others" or to none. That is not the case. Many people who belong to a certain group A will be able to relate fabulously to other members of group A and could be great teachers for them, while having a very poor ability to relate to members of another group B. In fact, we all feel more comfortable with and have a much better understanding of people who are similar to us - that is just human nature. But imagine that you are a community college administrator interviewing candidates for a math lecturer position, and along comes a member of group A who is able to convince you that he/she has a very good understanding of the challenges faced by members of group B and is really passionate about helping students from group B overcome those challenges and succeed. Wouldn't you be impressed and see great value in the possibility of employing such a person - at least assuming his/her general math lecturing credentials are just as good as the other candidates you're comparing them to?
Note that those sorts of qualities the colleges are looking for are of a much more specific kind that just someone having generally good people skills and a general ability to respect others. So yes, there is a lot "more to say" than the level of generality you are addressing.
With that said, of course having exceptionally good people skills can be impressive in its own right, and to some extent I believe you can tick off some of the right boxes without necessarily going down to the level of showing that you are passionate about helping very specific groups of people. Certainly showing a general respect for all people, whoever they may be, is a very good starting point for selling yourself in a job application.