Paraphrasing the discussion in Wooldrigde's introductory econometrics textbook, consider the following example:
It may make sense for the partial effect of the dependent variable with respect to an explanatory variable to depend on the magnitude of yet another
Example (not exactly continuous, but the logic is very similar for "truly" continuous variables):
price = \beta_0 + \beta_1 squarefeet +\beta_2 bedrooms + \beta_3 squarefeet\cdot bedrooms + \beta_5bathrooms+ u,
Now, the partial effect of $bedrooms$ on $price$ (holding other variables fixed) is
$$\beta_2 + \beta_3 squarefeet$$
If $\beta_3> 0$, then this implies that an additional bedroom yields a higher increase in housing price for larger houses. To assess the partial effect, evaluate the equation at interesting values of $squarefeet$, such as the mean value in the sample.
Note: The parameters on the original variables can become tricky to interpret. For example, the above equation shows that $\beta_2$ is the effect of $bedrooms$ on $price$ for a house with $squarefeet=0$...