How does the law of crosscutting relationships help scientists determine the relative age of rocks?

Because the fault that strikes through layers of rock must be younger than the rocks it strikes through.

So, let's imagine for a second that you have a 3-layered cake. Your 3 layers will consist of a chocolate cake, a vanilla cake, and a cheesecake, because you're crazy and your cake doesn't have to conform to standards. But mostly because cheesecake is great. On your plate, you put the chocolate cake on bottom, or the first layer. Then you put your vanilla cake in the middle. And finally, your cheesecake goes on top.

So, by definition, the oldest layer of cake would be chocolate, as it was put there first. Then vanilla and cheesecake are the second and third oldest, respectively. Now, because we've already established that this cake is pure insanity, you're going to cut out a big hole in the middle that pierces all 3 layers and stick in some new feature. Let's say... cookies.. yeah, cookies.

Now all of a sudden the big tower of cookies running through the middle of your cake is the youngest feature, as the other three layers had to be present originally in order for it to be there.

Now that I've gotten you all hungry for cake and cookies let's talk about rocks instead. This exact same concept can be applied to layers of rock underneath the surface. We know that we can relatively date layers of rock by knowing that the layers on bottom are older than the layers on top. So now we can say that any faults that run through these layers of rock would instantly become the new youngest layer, as the layers that were pierced by the fault had to be there first.

This picture sums it all up very well: