Zero Mean and Unit Variance
The questions of whether and why it's important, depends on the context.


For gradient boosted decision trees, for example, it is not important - these ML algorithms "don't care" about monotone transformations to the data; they just look for points to split it.
For linear predictors, for example, scaling can improve interpretability of the results. If you'd like to think of the magnitude of the coefficients as some indication of how much a feature is affecting outcome, then the features has to be scaled somehow to the same area.
For some predictors, in particular NNs, scaling, and in particular scaling to a particular range, can be important for technical reasons. Some of the layers use functions that effectively change only within some area (similar to the hyperbolic-family of functions), and if the features are too much out of the range, saturation can occur. If this happens, numerical derivatives will work badly, and the algorithm might not be able to converge to a good point.