They're right. If the null hypothesis is true, then the probability of getting any p value is equal. Put it another way, the p value is the probability of getting the observed data (or more extreme) under the null. So, under the null, 10% of the time you will get data with a p value of 10% or less; 20% of the time, you will get data with a p value of 20% or less; and so on. And that's the uniform distribution!

Here's an R example to play with:

  pvals <- replicate(10000, {
    x <- rnorm(100)
    y <- rnorm(100)
    t.test(x, y)$p.value
  })
  plot(density(pvals))

That will plot you a nice uniform line on [0, 1].

(NB: I have no idea why OP talked about p values following a normal distribution. That doesn't make sense to me, and I think the post has been deleted.)

> How is p uniformly distributed under H0?

That's the very definition of a p-value! The mapping of data to p-values is chosen to have a uniform distribution of p-values when the data is distributed according to the null hypothesis. That's the property that makes p-values interesting.