In explaining the meaning of regression coefficient I found that the following explanation very useful. Suppose we have the regression
Say $X$ changes by $\Delta X$ and $Y$ changes by $\Delta Y$. Since we have the linear relationship we have
$$Y+\Delta Y= a+ b(X+\Delta X)$$
Since $Y=a+bX$ we get that
$$\Delta Y = b \Delta X.$$
Having this is easy to see that if $b$ positive, then positive change in $X$ will result in positive change in $Y$. If $b$ is negative then positive change in $X$ will result in negative change in $Y$.
Note: I treated this question as a pedagogical one, i.e. provide simple explanation.
Note 2: As pointed out by @whuber this explanation has an important assumption that
the relationship holds for all possible values of $X$ and $Y$. In reality this is a very restricting assumption, on the other hand the the explanation is valid for small values of $\Delta X$, since Taylor theorem says that relationships which can be expressed as differentiable functions (and this is a reasonable assumption to make) are linear locally.