Understanding stratified cross-validation I read in Wikipedia: 

In stratified k-fold cross-validation, the folds are selected so that the mean response value is approximately equal in all the folds. In
  the case of a dichotomous classification, this means that each fold
  contains roughly the same proportions of the two types of class
  labels.



*

*Say we are using CV for estimating the performance of a predictor or estimator. What would mean response value (MRV) mean in this context? Just the average value of the predictor / estimator? 

*In what scenarios  would "achieving approximately the same MRV" in all folds be actually important? In other words, what are the consequences of not doing so?

 A: The mean response value is approximately equal in all the folds is another way of saying the proportion of each class in all the folds are approximately equal.
For example, we have a dataset with 80 class 0 records and 20 class 1 records. We may gain a mean response value of (80*0+20*1)/100 = 0.2 and we want 0.2 to be the mean response value of all folds. This is also a quick way in EDA to measure if the dataset given is imbalanced instead of counting.
A: Cross-validation article in Encyclopedia of Database Systems says:

Stratification is the process of rearranging the data as to ensure
  each fold is a good representative of the whole. For example in a
  binary classification problem where each class comprises 50% of the
  data, it is best to arrange the data such that in every fold, each
  class comprises around half the instances.

About the importance of the stratification, Kohavi (A study of cross-validation and bootstrap for accuracy estimation and model selection) concludes that:

stratification is generally a better scheme, both in terms of bias and  variance, when compared to regular cross-validation.

A: Stratification seeks to ensure that each fold is representative of all strata of the data. Generally this is done in a supervised way for classification and aims to ensure each class is (approximately) equally represented across each test fold (which are of course combined in a complementary way to form training folds).
The intuition behind this relates to the bias of most classification algorithms. They tend to weight each instance equally which means overrepresented classes get too much weight (e.g. optimizing F-measure, Accuracy or a complementary form of error).  Stratification is not so important for an algorithm that weights each class equally (e.g. optimizing Kappa, Informedness or ROC AUC) or according to a cost matrix (e.g. that is giving a value to each class correctly weighted and/or a cost to each way of misclassifying). See, e.g.
D. M. W. Powers (2014), What the F-measure doesn't measure: Features, Flaws, Fallacies and Fixes. http://arxiv.org/pdf/1503.06410
One specific issue that is important across even unbiased or balanced algorithms, is that they tend not to be able to learn or test a class that isn't represented at all in a fold, and furthermore even the case where only one of a class is represented in a fold doesn't allow generalization to performed resp. evaluated. However even this consideration isn't universal and for example doesn't apply so much to one-class learning, which tries to determine what is normal for an individual class, and effectively identifies outliers as being a different class, given that cross-validation is about determining statistics not generating a specific classifier.
On the other hand, supervised stratification compromises the technical purity of the evaluation as the labels of the test data shouldn't affect training, but in stratification are used in the selection of the training instances. Unsupervised stratification is also possible based on spreading similar data around looking only at the attributes of the data, not the true class. See, e.g.
https://doi.org/10.1016/S0004-3702(99)00094-6
N. A. Diamantidis, D. Karlis, E. A. Giakoumakis (1997),
Unsupervised stratification of cross-validation for accuracy estimation.
Stratification can also be applied to regression rather than classification, in which case like the unsupervised stratification, similarity rather than identity is used, but the supervised version uses the known true function value.
Further complications are rare classes and multilabel classification, where classifications are being done on multiple (independent) dimensions.  Here tuples of the true labels across all dimensions can be treated as classes for the purpose of cross-validation. However, not all combinations necessarily occur, and some combinations may be rare. Rare classes and rare combinations are a problem in that a class/combination that occurs at least once but less than K times (in K-CV) cannot be represented in all test folds. In such cases, one could instead consider a form of stratified boostrapping (sampling with replacement to generate a full size training fold with repetitions expected and 36.8% expected unselected for testing, with one instance of each class selected initially without replacement for the test fold).
Another approach to multilabel stratification is to try to stratify or bootstrap each class dimension separately without seeking to ensure representative selection of combinations. With L labels and N instances and Kkl instances of class k for label l, we can randomly choose (without replacement) from the corresponding set of labeled instances Dkl approximately N/LKkl instances. This does not ensure optimal balance but rather seeks balance heuristically. This can be improved by barring selection of labels at or over quota unless there is no choice (as some combinations do not occur or are rare).  Problems tend to mean either that there is too little data or that the dimensions are not independent.
A: This page of the documentation of scikit-learn has a pretty nice visual explanation of what are the differences between cross-validation sampling approaches. Here are some images for the methods you asked taken from the mentioned page.
As you can see, with KFold CV you divide the data in equal parts and pick train and test sets. For this method, I suggest you to include a sample shuffling process to avoid any eventual bias on this division.

For stratified KFold CV, you consider dividing train and test sets for each strata, since there is a imbalance on sample sizes. This is essential for classification problems, but you may consider using it when doing regression if you can divide data into clusters.

A: A quick and dirty explanation as follows:
Cross Validation: Splits the data into k "random" folds
Stratified Cross Valiadtion: Splits the data into k folds, making sure each fold is an appropriate representative of the original data. (class distribution, mean, variance, etc)
Example of 5 fold Cross Validation:

Example of 5 folds Stratified Cross Validation:

