What does the tilde (~) mean in my composer.json file?

Tilde means next significant release. In your case, it is equivalent to >= 2.0, < 3.0.

The full explanation is at Tilde Version Range docs page:

The ~ operator is best explained by example: ~1.2 is equivalent to >=1.2 <2.0.0, while ~1.2.3 is equivalent to >=1.2.3 <1.3.0.

Another way of looking at it is that using ~ specifies a minimum
version, but allows the last digit specified to go up.

Seldeak’s below comment is a simple sum up explanation of the Composer documentation.