Demonstrated interest is a term you'll hear often when people talk about college admissions. It means, well, exactly what it says: you've demonstrated that you're interested in a college you've applied to.
It seems like it should be obvious that you're interested if you've applied, but that's not necessarily the case. University admissions staff know that you may have applied because you really want to be there. They know that you may have applied because it's your safety school and not actually someplace you want to be if you can help it. They know that you may have applied because your boyfriend, girlfriend, or best friend applied, and you're actually kind of secretly hoping that you don't get in. They know that your family may have pressured you to apply. They know that you may actually have no idea why you applied--that happens all the time.