I standardized my x and y variables by subtracting by the mean and dividing by the standard error. I thought that was "fixing" my data, but I noticed the regression t-value that's produced in R is the same using either method. I'm just curious if anyone can explain the intuition behind that.
Are you familiar with a typical t test for the difference between a mean and a hypothesized mean? In that test you subtract the mean and divide by the standard deviation to get the T-value.
It's the same in this regression, when you don't standardize your x and y variables the regression t test is doing it for you. If you do standardize the x and y variables beforehand then you've set the mean to 0 and the standard deviation to 1, now when the regression T-value is calculated it involves subtracting 0 and dividing by 1 so it will stay the same.
Your data doesn't actually need "fixing" for linear regression. The fixing can help in cases where, for example, the mean and standard deviation are very large and the analysis would involve numbers larger than the computer can store. The fixing makes all the numbers a suitable size and avoids numerical instability.
The t-value gives the distance in units of standard error between the reference mean and the comparison mean. It does not matter what actual numbers you have, only the difference between those two numbers. If you change the units, the distance is the same, so your t-value and the area of the curve excluded by that t-value is the same.