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Statistical tests are not set up to prove that anything is true, rather they are set up to test the null hypothesis that some phenomena or event happened by chance, or that there are no differences in groups of subjects/events. They intend to disprove the null hypothesis.
A statistically significant result indicates evidence that the phenomena could not have happened by chance thus, we fail to reject the null hypothesis.
A statistically significant pearson product moment correlation is merely saying that it is highly unlikely that the obtained correlation happened by chance. However, as many people have commented, correlational results do not imply causation because of the potential for spurious variables.
For example, if one were to look at the correlation for per pupil expenditure money for each state in the US, and the academic achievement (SAT scores) of the students within each state, there is a inverse correlation between the amount per pupil expenditure and academic achievement (SAT scores) (More money = Less achievement). However, this correlation fails to take into account the percentage of students taking the SAT for each state. The states that spend the least per pupil have the fewest percentages of their students taking the exam and the states that spend the most per pupil have the highest percentages of students taking the SAT.
Maybe in some states they only want the smart kids to take the SAT, maybe they don't have the expectation for all kids to go to college, so they don't push the SAT on students with lower academic achievement. All of which are valid questions.
Therefore, after you statistically control for the percentage of students taking the SAT in each state, there is a positive correlation (as expected) between per pupil expenditure and SAT achievement.
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