I conducted a multiple regression analyses that included an interaction term (continuous x continuous). The interaction term was significant, but none of the simple slopes (1 SD above the mean, at the mean, 1 SD below the mean) were significant. How should I interpret this?
The interaction term is specifying that variable $X$ is being moderated by variable $Z$. A significant p-value suggests that the slope (or regression weight) for variable $X$ is significantly different for different values of $Z$ (e.g., -1 SD, M, and +1 SD).1 Said differently, the interaction "can be interpreted as the amount of change in the slope of the regression of $Y$ on $X$ when $Z$ changes by one unit" (source).
When evaluating the simple slopes of $X$, different values of your moderator variable $Z$ can be used to probe where the slopes might be different (as is common for continuous variables and used in your example: -1 SD, M, and +1 SD). Not finding that the slope of $X$ is significant for any of these values does not invalidate your interaction.
With simple slopes, you are testing whether the slope (or regression weight) of $X$ is different from zero at the value of $Z$. Therefore, you can have slopes of $X$ that are significantly different from one another but not be significantly different from zero.
Let's say that you have a significant interaction and the slope of $X$ at -1 SD for $Z$ is -0.100 and the slope of $X$ at +1 SD for $Z$ is +0.100. The slopes of $X$ can be significantly different from one another and they would indeed "cross paths" forming your interaction but the values of $\pm$0.100 themselves may not be significantly different from zero.
We often see that at least one slope in our simple slopes is different from zero, which provides an indication that that value of $Z$ is having the specific effect in the interaction (because that slope is more perpendicular to a horizontal line, which is a slope of zero). With non-significant simple slopes, the effect is less obviously one value of $Z$ or another and rather reflects the general moderation of $X$.
1 You can use any values for continuous variables but -1 SD, M, and +1 SD are most common. If you have theoretically important values, then it would be better to use those instead of this arbitrary standard.
Edit: To provide more information about what @Alexandre Gareau and @Pere said about cross-over interactions, which applies to this circumstance, here's a nice descriptive example: https://www.theanalysisfactor.com/interactions-main-effects-not-significant/
It could happen that you found an instance of cross-over interaction or a close related phenomenon, since the cross-over interaction concept is usually used in relation to ANOVA instead of linear regression. That's what @Alexandre Gareau said and agrees with the model proposed by @not_least_squares.
Anyway, since you said in comments that one main factor is significant when non taking interaction in account, I think your problem is multicollinearity. When you take in account main effects and interaction, you have three predictor variables ($X_1, X_2$ and $X_1 \times X_2$) and I suspect multicollinearity arises because $X_1 \times X_2$ is highly correlated with $X_1$ or $X_2$. Therefore, coefficient estimates may change erratically without changing the predictive power of the model nor the dependent variable estimates.
I'll give an example where I would expect to happen the same issues you have found. Let's imagine we have a sample of offers of apartments for sale in a given area, and we want to predict selling price from the following variables:
- $X_1$: usable floor area (in m2).
- $X_2$: floor to ceiling height (in m).
If we run a regression analysis with those variables, we are likely to find that usable floor area is highly significant while height isn't.
If we take in account an interaction, we will just be adding another predictor variable $X_1 \times X_2$, that is, inner volume of the apartments, and, since height is nearly constant, volume will be highly correlated with apartment usable floor area. The same predictions that can be made using floor area can be made using inner volume, there is nearly no advantage in using one or the other, and definitively no advantage on using both.
Therefore, if I were you, I would compare models with just the significant main effect against models with just the interaction and models with both. If there is not a big improvement in using one of the last two, I would just use the model with the significant main effect.
And just as a closing note: interactions are very useful when performing ANOVA and analysing planned experiments, but beware of interactions when using regression to analyse observational data.
You interpret it exactly as you'd think. Imagine the true model is
$$ Y = \beta_0 + \beta_1 X_1 X_2 + \varepsilon $$
So, the slope of $X_1$ depends on the value of $X_2$ and vica versa, and when one variable is zero, the other has no slope. Note that if you shifted the variables (e.g. mean-centering) in this model, the main effects would reappear.