Is the reason when demand rises and causes price to rise because there will temporarily be a shortage and that will result in a higher price?is...

That is basically correct.  Economic theory says that
increased demand will lead to a shortage and that firms will react by raising their
prices.

The way that this is likely to work in real life,
though, is that the owners of stores will look and see that this particular product is
flying off their shelves faster than they would have expected.  When this happens, they
will think that they might well be able to sell just as many at a higher
price.

But economically speaking, that is exactly what
happens -- there is a temporary shortage resulting in a higher
price.