It is needed to minimizes intra-cluster variance. You want to have in a given clusters those elements (data) that are most similar to the cluster center (or prototype) and one way of doing this is by minimizing the variance inside the cluster (intra-cluster). This can be combined with the opposite criterion of maximizing the difference between different cluster (inter-cluster), so another objective function coud be used together with the intra-cluster one.

Sir actually the cluster hypothesis is to maximizing intra-cluster similarity and minimizing inter cluster similarity. But you are saying minimizes intra-cluster varience. how it is?  

minimizing inter-cluster variance is "equivalent " to maximizing intra-cluster similarity because the less variance the more similarity among the data. As an extreme case if all the data points are exactly the same, the variance is 0 and the similarity is maximal (if the similarity measure is normalized it would have a value of 1).  Therefore there is no contradiction at all.

sorry! I made a mistake!! I meant "intra-cluster variance and not "inter-cluster variance", so the right text is:

minimizing intra-cluster variance is "equivalent " to maximizing intra-cluster similarity because the less variance the more similarity among the data. As an extreme case if all the data points are exactly the same, the variance is 0 and the similarity is maximal (if the similarity measure is normalized it would have a value of 1). Therefore there is no contradiction at all.

Role of objective function in FCM is to minimize the fuzzy compactness. 

Minimizing objective function means increasing similarity among all the components within an object and reducing similarity between components of one object with others.

To gain a practical sample about FCM and its objective function, you can refer to the following publication:  

Article Automatic building extraction in dense urban areas through G...

Does it mean that if objective function is minimize then the accuracy (confusion matrix) will be improved (maximize) 

Yes, optimizing the objective function means that the clustering obtained is better and therefore the accuracy should increase.

Yes, increasing similarity among all the components within an object and reducing similarity among different objects improve the final results. However, this improvement continues to a certain value. If the value of objective function is lower than this value, the final results will not change meaningfully. This value is determined via your application.  

The cluster mechanism rely on two steps which are: 1- selection 2- displacement. if the value of objective function high that means the data point far from the center point or cluster center, so the cluster well keep repeating these steps until it reaches the optimal value of objective function. That definitely should increase the accuracy