How to describe in words the logical conclusion of a one-tail t-test? Lets say we have the following null and alternative hypothesis:
$$H_0: \mu = \mu_0, \; H_A:\mu > \mu_0 $$
Thus, the p-value in this situation represents the probability of obtaining test results equal to or greater than the results actually observed (under the assumption that the null hypothesis is correct): $$P(t_{df = n-1} \geq t_{obs})$$
Finally, here is a visualization of the theoretical sampling distributions of the two hypotheses:

The red curve and green curves represents the sampling distributions of the null and alternative hypotheses, respectively. The blue vertical line represents the value of the observed results.
Now it is clear that the p-value for these observe results would be very close to 1. But I am struggling to logically explain the conclusion from this result (p-value) in words. Here is my horribly clunky attempt:
"Since the p-value of this t-test is so high, it means it is very probably that any future observations would have higher values than the value of the current observation. And since the mean of the null hypothesis is closer to the current observation than the mean of the alternative hypothesis, it is more likely the current observation came from the null hypothesis than the alternative hypothesis. Moreover, we cannot "accept" the null hypothesis here since it is quite clear the observation does not belong to the null hypothesis. Therefore, we simply just do not reject the null.
My first and second sentence in particular don't seem logically connected. In other words, how can I explain that this high p value "favors" the null over the alternative?

I'm aware I'm entering a minefield here since the correct explanation of p-values in plain English is tricky. Here's my take on explaining the result of a large p-value:
A large p-value indicates that the observed difference, or more precisely the observed value of the test statistics, is compatible with the assumptions of the null hypothesis.
Here compatible means just that and it shouldn't be confused with "supports the null" and it doesn't inform on the plausibility of one hypothesis or the other.
It may be worth keeping in mind that the p-value is a random variable (i.e. it depends on the sampled data) and if the null hypothesis holds the p-values follow a uniform distribution between 0 and 1. In other words, if $H_0$ holds large p-values are as likely to appear as small ones. I think this is counter-intuitive and easily overlooked.
Here's a simulated example to illustrate the behavior under the null:
p.values <- rep(NA, 10000) 
for(i in 1:length(p.values)) {
    x <- rnorm(n= 20, mean= 0, sd= 1)
    y <- rnorm(n= 20, mean= 0, sd= 1)
    res <- t.test(x, y)
    p.values[i] <- res$p.value
}
hist(p.values, breaks= 30)


