I think it's important to consider what regression is doing. Then the coefficient of determination makes sense.
Let's say that we collect some data on the heights of people. From our data, we find a mean of 5'2" with a middle range (Q1 to Q3) of 4'2" to 6'2". Given a new person, what height do you guess?
Depending on your application, mean may or may not be what you're looking for, but let's say it is, since OLS regression is predicting conditional means. You could guess 5'2" with a range of 4'2" to 6'2", but that's such a wide range! You kind of have no idea how tall this random subject will be. However, in the absence of any other knowledge, you know that the mean value gets the right answer on average, so you guess the mean. You get the answer wrong--by a lot.
However, you could have collected other information about the people whose heights you measured. I would expect height differences between men and women. Certainly I expect height differences at different ages.
Now you know that the person whose height you have to guess is a 40-year-old man. You go look at your subjects with known height and don't have any 40-year-old men, but you have men who are 39, 41, and 42, who average 5'9" with a range of 5'8" to 5'10". You have much more confidence in your answer of 5'9". In fact, the subject was 5'8". By doing regression on the age and gender, you have decreased your error from 6" to 1".
The goal of regression is to decrease that variability by using other information. That other information form the predictive variables in your model. Regression then predicts the mean of a conditional distribution, conditioned on that other information (e.g. male and 40 years old).
Let's get back to the formula for the coefficient of determination. The total sum of squares is by how much you miss the correct value by guessing the average of all observations. The sum of squared errors is by how much you miss the right answer by predicting the mean of the conditional distribution. The sum squared of the regression is by how much you decrease your error by accounting for the additional information. Instead of naively guessing the overall mean, you something about the subject and tighten up your guess.
When we say that the coefficient of determination is SSReg/SSTotal, we are saying what percentage of the variability in the observations is accounted for by the additional information (predictive variables). If we have a conditional distribution with little variability, then we can make a very tight guess about what a new observation would be.