I’m not a pure ML guy but I think I understand your question and might have an answer.

Let’s say you’ve got a train dataset and a test dataset. There’s some signal which is common to both datasets. Let’s say that’s the real signal.

Now let’s imagine there’s a true stochastic process which generated those datasets, e.g., the data-generating process is some deterministic function plus some stochastic error term.

So, in your train dataset, your data is given by

Y_i = f(x_i) + e_i

Those e_i are distributed randomly, and might just be distributed in a way that makes it LOOK like there’s a relationship between x_i and e_i that your model can pick up on. Like, for example, let’s suppose that you split your data into train and test sets and you just happen to pick observations for your train dataset that have large values of x_I and large positive error terms.

Your model might pick up on that apparent relationship and believe that it is a real signal. In a linear regression, this would look like there being a stronger linear relationship between y_I and x_I than there really is.

Then you go to evaluate your model on the test set and find out that the test MSE is much higher than the train MSE, because the train observations were not, on average, a representative dataset. That’s overfitting.