Ballistic missile defense is a pretty different problem from defending against aircraft. While both involve defending from stuff above you, they take radically different paths at radically different speeds. An aircraft is moving at roughly the speed of sound (a bit under for most stuff, or maybe somewhat over for some of the fancier bombers) on an unpredictable path at an altitude between maybe 40,000ft and 100ft. An ICBM warhead is traveling at maybe mach 25, is coming down on a near vertical trajectory from a couple thousand miles up, and is on an entirely predictable trajectory.

The differences make for extremely different requirements. To shoot aircraft down, you want a missile that can travel somewhat faster than the aircraft. By not going too fast, they can stay maneuverable, to counter the aircraft's own maneuverability. You can use a proximity fuze so that once you get somewhat close to the target, the missile explodes and destroys the aircraft with shrapnel. You typically have some hours of warning to get the missile into position.

To shoot down an ICBM warhead, you need a much faster missile in order to perform the intercept at a useful altitude. You have maybe 20 minutes of warning at best, and probably no more than 5-10 minutes knowing the exact target. The target doesn't maneuver so the missile doesn't need to counter that. The target is maneuvering so fast, however, that a proximity fuze does you no good if you're using chemical explosives. The target is moving much faster than your explosives explode, so timing the explosion to hit the target with your shrapnel field is too hard. One option is to hit the target directly (kinetic kill), which means hitting something with a cross section of a couple of square feet while your closing velocity is several miles per second. This is hard. Another option is to use nuclear warheads on the interceptor. The detonation won't physically destroy the incoming warheads, but the radiation flux can destroy their electronics.

The answer to "how do you defend against ICBMs?" is more or less: you don't. No practical system to defend against a large-scale ICBM attack has ever been developed, or even planned. Defending a few targets at great expense is possible. The Soviets had their Galosh missiles defending Moscow, which are still in place. The Americans built the Safeguard system to defend ICBM silos around North Dakota, but it was decommissioned quickly because it just cost too much. The Americans now have the Ground-Based Midcourse Defense in Alaska and California. This can defend much of the country against ICBMs, but only a small number of them, so the intent is to prevent e.g. North Korea from carrying out a small strike, not stopping a full-scale Soviet/Russian/Chinese strike. Some Aegis destroyers operated by the US Navy also have the ability to shoot down ICBMs, but again the numbers are limited. In general it's cheaper to build an ICBM than to build something that can intercept an ICBM (especially since one ICBM can carry 10 or more warheads, all of which need to be shot down separately unless you can somehow destroy the missile while it's still launching), so unless you have great economic superiority over your adversary, they can always out-build your defenses.