There are multiple strategies, and a sim rarely models more than one or two.
A missile can, first and foremost, destroy the target by direct impact and detonation within/on the target.
There exist SOME missiles (SAMs specifically) which will get in the way of the target and explode in front of it, creating a shrapnel cloud for the target to fly through. This is what the NIKE missile used to do, but this is no longer employed in modern systems, AFAIK - at least for the most part.
Most AAMs and SAMs (but not all) feature a proximity fuze. IF the missile misses, it can still sense that it is close enough to the target with a Target Detection Device (also called the proximity Fuze, and it is a separate sensor from the missile's seeker) and detonate its warhead. Modern missiles can time the detonation to get the best effect out of it.
Some prox fuzes are laser, some radio, and there have been magnetic influence ones as well.
Notable and interesting exceptions are the StarStreak, which launches several projectiles after boosting.
Missiles like the Stinger are typicall hit-to-kill, and they don't have a proximity fuze (they are small, cheap missiles with no space for all the toys).
Most games will model hit-to-kill and a simple proximity fuze, nothing else.