The k-nearest neighbor algorithm relies on majority voting based on class membership of 'k' nearest samples for a given test point. The nearness of samples is typically based on Euclidean distance.
Consider a simple two class classification problem, where a Class 1 sample is chosen (black) along with it's 10-nearest neighbors (filled green). In the first figure, data is not normalized, whereas in the second one it is.
Notice, how without normalization, all the nearest neighbors are aligned in the direction of the axis with the smaller range, i.e. $x_1$ leading to incorrect classification.
Normalization solves this problem!