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?

You do not give data to go with your question, so I will
illustrate in terms of fictitious data, sampled in R, that may be suitable for illustration.
set.seed(2022)
x = rnorm(100, 50, 10)
summary(x);  length(x);  sd(x)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  20.99   45.30   51.72   51.39   58.66   78.87 
[1] 100         # sample size
[1] 10.20093    # sample standard deviation.

Here is a stripchart that shows the 100 observations.
stripchart(x, pch="|")


If you want to test $H_0: \mu \le 60$ against $H_a: \mu > 60.$
The corresponding test is a right-sided one-sample t test.
The test statistic is $T = \frac{\bar X = \mu_0}{S/\sqrt{n}}
= \frac{51.387 - 60}{10.20/\sqrt{100}} = -8.4429.$
From a printed table of Student's t distributions, you can
find (at least approximately) that the critical value of the test is $c = 1.6604.$ You would look on row DF=99 of the table. if there is one,
otherwise find the nearest. This is the value that cuts 5% of the area from the upper tail of Student's t distribution with 99 degrees of freedom. [The exact value below is from R.]
qt(.95, 99)
[1] 1.660391

This means that you would reject $H_0$ in favor of $H_q$ if $T \ge c = 1.6604.$ However, in fact, the statistic $T = -8.4429$ from your data
is (much) too small to reject $H_0.$
In R, the test looks like this:
t.test(x, mu = 60, alt="greater")

        One Sample t-test

data:  x
t = -8.4429, df = 99, p-value = 1
alternative hypothesis: 
 true mean is greater than 60
95 percent confidence interval:
  49.69371      Inf
sample estimates:
mean of x 
 51.38746 

It seems that the alternative hypothesis was set up expecting
an alternative population mean $\mu_a$ that may be greater than $60;$ instead the population mean appears to be smaller.
The P-value
is the probability under the density curve for
Student's t distribution with 99 degrees of freedom that lies above $-8.4429.$ This probability is
very nearly $1,$ the P-value of the test.
The graph you pasted into your question seems to be for a
normal test (z test) instead of a t test [and with a specific alternative (green) in mind]. The plot below shows
the density function of $\mathsf{T}(99)$ in maroon, the observed
value of the $T$ statistic (heavy blue vertical line), and
the critical value $c = 1.6604$ (vertical dotted line).

R code for figure:
hdr="Density of T(99)"
curve(dt(x, 99), -9, 5, col="maroon", lwd=2, main=hdr)
 abline(v = -8.4429, col="blue", lwd=2)
 abline(v = 1.6604, col="darkgreen", lwd=2, lty="dotted")
 abline(v=0, col="green2")
 abline(h=0, col="green2")

