I am not sure why this is so unclear. The null hypothesis of "no treatment effect" is equivalent to saying "both groups were sampled from the same distribution" which is equivalent to "the means of the treatment group and control groups are exactly equal"
The p-value is the probability of getting as or more extreme of the difference between sample means that you observed assuming that there is in reality no difference in population means. If the treatment and control groups are different at baseline (or become different over the course of the study for reasons other than the treatment) then this assumption is false.
What you quoted doesn't say what you just said. "both groups were sampled from the same distribution" is not the same as "the means of the treatment group and control groups are exactly equal." The *expected* means of the groups is equal. The math behind the statistical models takes into account random group assignment and so there is no expectation that the groups start exactly the same (or change in similar ways over the course of the experiment).