How do standardization and normalization impact the coefficients of linear models?
When you have a linear regression (without any scaling, just plain numbers) and you have a model with one explanatory variable $x$ and coefficients $\beta_0=0$ and $\beta_1=1$, then you essentially have a (estimated) function:
$$y = 0 + 1x .$$
This tells you that when $x$ goes up (down) by one unit, $y$ goes up (down) by one unit. In this case it is just a linear function with slope 1.
Now when you scale $x$ (the plain numbers) like:
scale(c(1,2,3,4,5))
           [,1]
[1,] -1.2649111
[2,] -0.6324555
[3,]  0.0000000
[4,]  0.6324555
[5,]  1.2649111

you essentially have different units or a different scale (with mean=0, sd=1).
However, the way OLS works will be the same, it still tells you &quot;if $x$ goes up (down) by one unit, $y$ will change by $\beta_1$ units. So in this case (given a different scale of $x$), $\beta_1$ will be different.
The interpretation here would be &quot;if $x$ changes by one standard deviation...&quot;. This is very handy when you have several $x$ with different units. When you standardise all the different units, you make them comparable to some extent. I.e. the $\beta$ coefficients of your regression will be compareable in terms of how strong the variables impact on $y$ is. This is sometimes calles Beta-Coefficients or Standardised Coefficients.
A very similar thing happens when you normalise. In this case you will also change the scale of $x$, so the way how $x$ is measured.
Also see this handout.