The statistical concept of 'significant' vs. 'not significant' can be understood by comparing to the legal concept of 'guilty' vs. 'not guilty'.
In the American legal system (and much of the world) a criminal defendant is presumed innocent until proven guilty. If the evidence proves the defendant guilty beyond a reasonable doubt, the verdict is 'guilty'. Otherwise the verdict is 'not guilty'. In some countries, this verdict is 'not proven', which is a better description. A 'not guilty' verdict does not mean the judge or jury concluded that the defendant is innocent -- it just means that the evidence was not strong enough to persuade the judge or jury that the defendant was guilty.
In statistical hypothesis testing, you start with the null hypothesis (usually that there is no difference between groups). If the evidence produces a small enough P value, you reject that null hypothesis, and conclude that the difference is real. If the P value is higher than your threshold (usually 0.05), you don't reject the null hypothesis. This doesn't mean the evidence convinced you that the treatment had no effect, only that the evidence was not persuasive enough to convince you that there is an effect.