This is something which is pretty well discussed in chapter 8 of John Fox's book, Applied Regression Analysis and Generalized Linear Models, or Weisberg's Applied Linear Regression. Both emphasize that your question is related to Nelder's (1977) principle of marginality.
From this last book for example:
The approach to testing we adopt in this book follows from the marginality
principle suggested by Nelder (1977). A lower-order term, such as the A main
effect, is never tested in models that include any of its higher-order relatives
like A:B, A:C, or A:B:C. [...]
An analysis of variance table derived under the marginality principle has
the unfortunate name of Type II analysis of variance. [...]
Type III analysis of variance violates the marginality principle. It computes the test for every regressor adjusted for every other regressor; so,
for example, the test for the A main effect would include the interactions
A:B, A:C, and A:B:C.
The key point is that, with "type II" ANOVA, the $F$-tests based on the sum of squares used in this decomposition are valid (i.e., do really test main effects) only when interaction is absent.
Type III ANOVA allows for testing main effects in all cases, but do ask a different research question, and should not be used carelessly.
As an intuitive answer however, the idea of non interpreting main effects when interaction terms are significant could be the following: if A:B is significant, then both A and B do play an important role in the process. Furthermore, in many instances where we can observe complex interaction patterns, asking for main effects of A and B can be simply meaningless, since the expression of A depends too much on the expression of B. (For example, let's imagine a fertilizer that would increase yields only on very wet soils, but that would drastically decrease yields on dry soils. There would be a strong interaction fertilizer:irrigation, but it would be tricky to talk about the "main effect" of this fertilizer: this simply depends too much on watering.)