Interpreting effects of linear model In Alan Agresti's book, Foundations of Linear and Generalized Linear Models (section 1.2.3), letting $y_i$ is a college student's math test score, $x_{i1}$ the number of years of math education, $\mu_i = E(y_i) = \beta_0+\beta_1x_{i1}$, he writes "...we might say 'if this model holds, a one-year increase in math education corresponds to change of $\beta_1$ in the expected math achievement test score.' However, this may suggest the inappropriate casual conclusion that if a student attains another year of math education, her or his math achievement test score is expected to change by $\beta_1$."
He writes that this conclusion is not valid because "a higher mean test score at a higher math education level (if $\beta_1 >0$) could at least partly reflect the correlation of several other variables with both test score and math education level....".
I'm not sure why the first statement is wrong. If the model holds, then my understanding is that we know that $E(y_i) = \beta_0+\beta_1x_{i1}$ holds, hence it doesn't matter what the correlation is at other levels because the equation suggests that the first statement holds.
He write that a more appropriate interpretation is "If the model holds, when we compare the subpopulation of students having a certain number of years of math education with the subpopulation having one fewer year of math education, the difference in the means o their math achievement test scores is $\beta_1$."
Why is this interpretation more appropriate? More specifically, why is he using subpopulation, and why is he switching to a difference in means, rather than sticking to one mean?
 A: This is best explained by considering what object you're really estimating and viewing this problem in a causal framework.
Let's first consider what you're really modeling.
Let me write your model on an individual level as $$y_i = \beta_0 + \beta_1 x_i + \epsilon_i \,,$$where I introduce an error term $\epsilon_i$, which is supposed to capture everything that is unexplained by the model.
To give the model any meaning, as in your question, we have to impose restrictions on the errors. Most commonly we assume $\mathbb{E}[\epsilon_i \mid x_i=x] = 0$. Then the we get the same model for the conditional mean as in your question$$\mu(x) = \mathbb{E}[y_i \mid x_i=x] = \beta_0 + \beta_1x\,.$$
If you now consider $\mu(x')$ at some point $x'$, then this object corresponds to the mean of the outcomes $y_i$ of the subpopulation where $x_i = x'$. Hence, we may write $\beta_1 = \mu(x + 1) - \mu(x)$, but this does not mean that the individual outcomes increase (or decrease) by $\beta_1$ but only the conditional means, i.e. the means of the subpopulations are different by $\beta_1$.
Secondly, without getting technical, assume that there is another variable influencing $y_i$ and $x_i$. In your setting this could be mathematical intelligence. Individuals who have a higher mathematical intelligence score better on math tests (on average) but are also more likely to have a longer math education (on average). If you then compare individuals who have a one-year longer education, it is not clear if their better performance on the math test is due to the increase in education alone, or also due to their higher affinity towards math which led to the longer education. Hence, you cannot simply interpret $\beta_1$ as the effect of one-year more education on the individual alone, but only as a description of the subpopulations, i.e. the conditional mean, as seen above. For reference see omitted-variable bias and endogeneity.
Lastly, it is possible to get causal interpretations like the ones you aim for; however, you have to make causal assumptions for that. One way to do this is to describe how each observed and unobserved variable is related in your model. I suggest reading up on causal graphs.
