Small general-aviation aircraft often use reciprocating (piston) engines, and piston engines are notoriously unreliable compared to gas-turbine (jet) engines. As a result, a private pilot who flies long enough is likely to experience engine failures, although this isn't really universal.
In fact, commercial aviation is what it is today in part because of the move to jet engines. Prior to the use of jet engines, the giant piston engines on airliners were a constant source of problems, and it was not unusual for an engine to fail in flight. Jet engines, in contrast, are so reliable today that many airline captains will never experience a mechanical failure of a jet engine outside of a simulator.
Engine failures do not necessarily lead to crashes. If you have a flat place to land, you can easily survive an engine failure (even on a single-engine airplane) without any harm to yourself and with little or no damage to the aircraft. After all, airplanes glide quite well even without power, so dealing with total engine failure is essentially just a matter of finding a place to land promptly. On multi-engine aircraft, the failure of a single engine is even less likely to cause a crash; it typically just means an emergency landing at the nearest suitable airport.
All pilots are taught how to deal with engine failure, because it's important to do the right things quickly when an engine failure occurs. Provided that a pilot does what he needs to do, engine failure usually ends well for the pilot and for the airplane. US Airways Flight 1549 is a classic example of a total engine failure that was correctly handled by expert pilots. Walking away from an engine failure is the rule rather than the exception.
Of course, you might be saying that there are surely situations in which an engine failure leads to a crash, and that's true. If you're flying over jagged mountains and you lose all engine power, you probably won't survive. If you're flying over water and you lose all engine power, things might not go well unless the surface of the water is very smooth and you're close to land. There are certainly extra-prudent pilots who won't fly over mountainous terrain or water in single-engine piston aircraft, and that's their prerogative. Nothing forces you to fly in such situations, either, if you're uncomfortable about engine failure. But if you're in Iowa flying over hundreds of miles of flat farmland, an engine failure isn't going to be any big deal.
When accidents occur after an engine failure, it's usually not the engine failure that does it, it's the pilot's failure to properly handle the engine failure that does it. So if you are a well-trained and disciplined pilot, the risk is very low.