Hard time interpreting r output, p value confusion? 
I think I'm overthinking it, but this p-value for the exp.c:anx.c interaction (3.84e-09) is significant, right? In other words, it's much smaller than 0.001? Irrational numbers are confusing for me, and I also have a negative t-value that is throwing me off. 
I'm also confused at the bottom where it shows: p-value: < 2.2e-16, what does this mean? Overall for the model?
 A: 
p-value for the exp.c:anx.c interaction (3.84e-09) is significant, right? 

Yes, at the typical $\alpha=0.05$ level (i.e. $p=3.8 \times 10^{-9} \ll 0.05$) ($\ll$ means "much less than")

In other words, it's much smaller than 0.001? 

Not sure what you mean here.  Do you mean you want to use a cutoff of $\alpha=0.001$?  If so, then yes, $3.8 \times 10^{-9} \ll 0.001 $.

Irrational numbers are confusing for me

Not sure what you mean here. Mathematically, an "irrational" number is something like $\sqrt{2}$, a number that can't be represented as a ratio of whole numbers. Maybe you're confused by the 3.84e09 notation, where AeB stands for "A times 10 raised to the power of B" (?)

and I also have a negative t-value that is throwing me off.

A negative t-statistic means that the estimated parameter value is that many standard errors below zero. There's no difference between positive and negative t-statistics for standard two-tailed significance tests.

I'm also confused at the bottom where it shows: p-value: < 2.2e-16, what does this mean? Overall for the model?

Yes, that's a p-value for the full model vs. the null hypothesis of a constant model with $\hat y$ equal to the mean of the response variable.
