If the assumptions are not holding in the data, we will not be able to make correct decision. Because based on assuption(normality and homogentiy) only we create acception/rejection limit for F statistics in ANOVA.
If any of the assumptions are not present, then we will not be able to accurately calculate the acception/rejection limits ( ie. We can not calculate the actual P value accurately).
We make the assumptions of equality of variances and normal distributions, so that when we compare two groups, we know all else is the same. Therefore, they're only way left two samples are left to differ are in the Mean, and nothing else.