Why is it called 'multi-level thresholding', if only the best threshold is selected?
Using k1 and k2 you can calculate a between class variance value using the Otsu formula. However, there may be many combinations of k1 and k2 for which you have to calculate between class variance values to see which of them gives the highest between class variance value. That's what 'max' means. So whichever k1 and k2 gives you the highest between class variance, you choose those values of k1 and k2 as your thresholds. Same can be done if you want more thresholds.