People who are demonstrative easily and clearly show their emotions. A demonstrative person might shout "Hooray" and jump for joy at good news. A non-demonstrative person might feel no less excited, but refrain from demonstrating it.
To demonstrate means to show, so think of demonstrative as showing. In legal terms, demonstrative is used to describe evidence that shows that something happened––a note that says "I did it" might show, or demonstrate, an accused person's guilt. In grammar, demonstrative pronouns––this, that, these, those––indicate the thing or person that is being pointed out, or shown, as in “Officer, it was ‘that’ man who stole my purse!”