What do the neutral and ground connectors in outlets connect to?
Neutrals, grounds, NEMA 10, NEMA 14, and the First Rule of Electricity

The old way (3-wire) is called NEMA 10, and abuses the neutral as a protective chassis ground.  This works because the neutrals and grounds are bonded together in the main panel -- the purpose of a ground wire for an appliance is to send wayward current from the case back where it came i.e. the service entrance, while the neutral wire is going in the same direction, so to speak.  However, the NEMA 10 setup is problematic because if the combined neutral/ground breaks, the dryer's chassis will then float off to some shocking potential.

NEMA 14 (4-wire) eliminates this problem by separating the neutral and the ground, making them take two different paths back to that main panel bond.  That way, if the neutral wire to the dryer breaks or otherwise fails, the ground's still there to protect you from getting zapped.

When all else fails, remember the First Rule of Electricity: CURRENT FLOWS IN LOOPS!!! This means that electricity isn't ever trying to go to "ground", whatever-the-@$(!@$ that means, or to some random water pipe somewhere that really only is tangentially related to the entire electrical system.  It's always trying to get back to where it came from -- your electrical service from the utility or the battery in your car, for example.

Think of it like some district steam plant -- the utility is only loaning you the electrons or water molecules, as they are mere carriers of the energy you're getting sold.  Just like the steam utility would cut off someone who took all their valuable water and let it out into the atmosphere, the electric utility wants their electrons back just as much, if not more, and the same holds true for your car battery.