How to run experiments in such a way to understand the underlying sources of variation? Suppose I have some outcome variable $Y$ that depends on several factors $F_1, \ldots, F_m$, which I assume to be binary for simplicity. Each time I conduct an experiment on levels of the factors, I get a result of $Y$. I would like to understand how much of the sampling variation on $Y$ is attributable to each of $F_1, \ldots, F_m$. For example, 20% of variation when I sample $Y$ is due to $F_1$, etc. What are some general methods for inferring the variance that can be attributable to various underlying factors?

It might be helpful to construct an orthogonal design (https://support.minitab.com/en-us/minitab/18/help-and-how-to/modeling-statistics/doe/supporting-topics/basics/orthogonal-designs/). An orthogonal design will allow you to attribute a "proportion of variance explained" for each predictor (How to split r-squared between predictor variables in multiple regression?) which sums to the full model r squared.
