I am familiar with how to obtain the p-value from the t-statistic -
the t-statistic is calculated by looking at the difference in means in two samples. Then, one has to integrate over a t-distribution from -t-statistic to +t-statistic and if the area under the curve is >0.95, then one can reject the null hypothesis.

How does this translate for, say, the Shapiro-Wilk Test? What does one "do" with the W-statistic to get the p-value? There's no "W-distribution" or anything to integrate over.

I have the following data = {1.75, 3.48, 3.79, 0.32, 5.29, 3.31, 3.92, 0.66, 2.82}

Mathematica provides me with the following test statistics and p-values:
Cramer-von Mises
Shapiro Wilk
Watson U squared

but I do not know how Mathematica is doing this. I'm just trying to assess what type of distribution fits to my data.

Also, Mathematica determines that the Watson U squared test is the best dest to not reject the null hypothesis at the five percent level. Why did Mathematica determine that the Watson U squared test is the best?

I know R and Mathematica and Matlab have packages, but they don't tell you what they're actually doing. Wikipedia tells one how to determine the test statistic, but doesn't tell you what to do with it.