Question: Why is it that whenever witnesses are required in Islam, two women are required to take the place of only one man?

The Quran in Sura 2:282 says:

> 
> And let two men from among you bear witness to all such documents
>  [contracts of loans without interest]. But if two men be not
>  available, there should be one man and two women to bear witness so
>  that if one of the women forgets (anything), the other may remind her.
> 
> 
>

Answer: 
Allah has set a social system in Quran. According to this social system, women do not have to work, they must be fed and kept safe by their families or by the government. The women are not to be independent in the Islamic social system (I don't mean that they are prohibited to be independent, they don't have to. They have right to request shelter from government. They don't need to work. But they can if they want to, it is not prohibited.).

So, in the Islamic social system, women always depends to some others in both economic means and in emotional means. When a person (not necessarily to be woman) is dependent on someone else, he/she starts to think, live and behave as the one he/she depends to. This is a psychological fact.

So, if we consider the worst conditions, a woman witness can be threaten by her family or husband, or just she can feel herself under pressure of the one she depends to. She can hide some of the facts even if she doesn't want to do so.

Of course, there are also different reasons too.  

Women are too emotional. They usually exaggerate events. They have a passive role in the society (it is the same in the entire world). They can't easily express their feelings. They more try to take attention compared to men. They actually feel themselves dependent to other people.

Because of all these reasons (and maybe there are more reasons, Allah knows), two women's witness is not enough for clarifying a crime or some other matter. That doesn't mean `1 man = 2 women`, that's just to make the evidence more robust.
I think this area is heavily disputed (on witnesses in general). While the Qu'arn is very clear in your example, it does not explicitly extend that beyond financial matters.

The most commonly accepted reason that I have seen is that financial matters are generally not in the women's sphere of expertise or interests. On the contrary, many scholars say that a woman's testimony alone is fine on matters that deal with womanly areas, such as child birth. The number of women can be different depending on the scholar though.

Also, as a wrench in the system, in cases where a husband accuses his wife of unchastity, her testimony will invalidate his.