I don't have that particular gun, but the crossbolt safeties on lever guns I am familiar with are simply rods with a notch the width of the hammer cut out. When the notch is lined up with hammer, the hammer can fall into the notch slightly. This allows the falling hammer to make it all the way to the firing pin. When you activate the crossbolt safety, the notch isn't lined up with the hammer, so the hammer hits the crossbolt, thereby preventing it from reaching the firing pin. Other than that, it has no effect on the action.
Funny thing happened to me when my Marlin came back from the factory. Like most people, I never use the crossbolt safety, except in rare cases when racking rounds out of the gun. Anyway, I took it out to test it, and it would not fire. I examined the cartridges and could not see any evidence of a primer strike. I thought, "Is my firing pin broken? From the factory?!" Then I remembered the damned thing has a crossbolt safety, which they were kind enough to turn on for me. Doh! I guess it's probably a little better to dry fire with the crossbolt safety on, since the firing pin won't receive any wear.