Q:

Why should "are" be used here instead of "is"?

The sentence given in our book is as follows, 

More important than winning is developing the ability to work with others and developing leadership skills

We are supposed to find where the mistake is in the sentence... 4 options are given ( which I have made bold ) and one of them is correct ( meaning the mistake is there ).. 
As per the key given in our book, the correct answer is B option i.e. "is" and the key says that "are" should be used here... I am confused and can't understand why should "are" be used here? Anyone who can help me here, please? 

A:

Option B is the correct answer, as your answer key says. The error is in the use of the singular form of the copula "is" with a plural subject.
The correct form of the copula is "are" because the sentence in the non-inverted form is:

Developing the ability to work with others and developing
  leadership skills are more important than winning. 

This is a sentence with a compound subject. That means two subjects, so the plural is used.
Same pattern as:
Trees and bushes are important in gardens.
For clarification, here is the same sentence with the corrected word and with brackets added to contain each subject:
[I returned here to edit brace to brackets and cartouche to contain. I cannot believe I would have used the terms braces and cartouche here.

More important than winning are { developing the ability to work with others } and { developing leadership skills }

A:

Invert the sentence and it should be apparent:

Developing the ability to work with others and developing leadership skills are more important than winning.

Keep this trick in mind for the future.  It may be useful again.
That being said, this kind of ambiguity is a good indication that the sentence might be poorly written.  Most native speakers would naturally say is by default, because the "A is more X than B" structure is frequently used to compare two equal things.

More important than winning the race is striving to always do your best.
More enjoyable than eating the food is learning to prepare it for yourself.
More important than winning is learning to work together, and also developing leadership skills.

Keep in mind that there may be ways to write these sentences with better style.  Test questions like these are purposely designed to be difficult, while good writing is supposed to be easy to read.

A:

The is version sounds better than it should because of the complexity of the following subjects. You are somewhat led to believe that the first developing wraps up the remainder of the sentence into a single subject.
The answer becomes more obvious if you simplify the two subjects, so say:
developing the ability to work with others => cooperating
developing leadership skills => leading

More important than winning is cooperating and leading.

vs.

More important than winning are cooperating and leading.

