Recently, I learnt that one can also calculate a p-value for the Pearson correlation coefficient. My question is not about computation of this p-value. (Based on what I read from other posts, we first calculate the t-statistic and then convert it to the p-value.) I would like to ask about the meaning of this value because from what I know, the value of the correlation coefficient already tells us the strength of the correlation. Suppose that based on sample data for variables X and Y, we find that they are strongly positively correlated with the value of r close to 1. Then, what new information does the p-value I calculate for this coefficient tell me? I'm not looking for an answer that just tells me "p-value tells us how significant the correlation is" because I'm not really able to distinguish between "strength of the correlation" and "significance of the correlation". To any lay person, probably they would think that they mean the same thing.
Specifically, I'm referring to the p-values calculated using this formula:
$$ t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$$