How is the relationship between two variables $X$ and $Y$ supposed to "explain" $R^2\text%$ of the variation of the data? Suppose we have a linear regression and we calculate $R^2 = 0.81$. What do we mean when we say "the relationship between two variables $X$ and $Y$ explains $81\text%$ of the variation of the data"? Especially this word "explains" confuses me, since the linear regression just reduces the variance in the observed data $Y$ by $81\text%$. It doesn't explain anything.

Say you fit the model:
$$
Y_i=\beta_0+\beta_1X_i
$$
and you get:
$$
R^2=0.81
$$
This means that your independent variable $X$ accounts for 81% of the variability in your dependent variable $Y$. Or, in other words, 81% of the variability in $Y$ is explained by the variability in $X$.
This is easy to see if you look at how $R^2$ is calculated:
$$
R^2=1-\frac{SS_{residuals}}{SS_{total}}
$$
where $SS_{total}$ is the total sum of squares, which represents the total variability of your data (in particular, the total variability of your dependent variable $Y$); and $SS_{residuals}$ is the residual sum of squares, which is a measure of the discrepancy between your data and the model. You can see how the lower $SS_{residuals}$, the better i.e. the higher $R^2$ will be.
Finally, recall that the total sum of squares can be partitioned as follows:
$$
SS_{total}=SS_{regression}+SS_{residuals}
$$
You can see how $SS_{residuals}$ represents the variability that's left after you remove the variability explained by the model, because $SS_{residuals}=SS_{total}-SS_{regression}$.
Hence, $\frac{SS_{residuals}}{SS_{total}}$ is the proportion of the total variability that isn't explained by the model, and therefore $1-\frac{SS_{residuals}}{SS_{total}}$ is the proportion of the total variability that is explained by the model.
