How often do we see normally distributed data
Your confusion is apt. Normally distributed data doesn't show up that often. Most of the real world datasets are more complex than normal. Many of natural occurring phenomenon (think: height of people in certain population) might be normal. But most of the cases where human behaviour plays a strong role (donations as you mentioned, incomes, peoples preferences) will show other distributions like fat tailed distributions or power law distributions.

But the result that you highlighted talks about a result in statistics called central limit theorem, which states that the mean that you infer from averaging will be normally distributed (irrespective of distribution of data). I'll explain below with an example.

Imagine you want to talk about heights of all males in United States. The first question that you might ask about such data is what's the central tendency (mean). But you may not have data about every male in the United States (getting that data will be too expensive). So you take a sample of let us say 100 people from each state. There's no way for you too know the distribution or shape of heights the entire population (you haven't collected that data!). You take the sample that you collected and calculate average of the heights. Can you make some statements about the how close is the average to the real mean of entire population? The computed average after sampling is a random variable since you will get different answer for every different sample. Central limit theorem says that this random variable is normally distributed with its mean same as the population mean and your estimate will get pretty closer as you have bigger samples. (Central limit theorem works in case of i.i.d. samples, that is samples are independent of each other and they are picked from same population.)

So in summary, normal distribution shows up more often with statistical tests because you are talking about distribution of average of sampled data, not the actual distribution of data (which can be non-normal). In a way, central limit theorem is why most of real world applications of statistics from simulations to election studies (psephology) works!

All the theory above answers the initial part of your answer. But for specifically running ANOVA test, you need the data to be (close) to normal distribution. Look at histogram of the data (yes, I know you have done sampling only once) and you can run tests for normality assumptions.