Why don't event horizons go out to the edge of the black hole's effective gravitational field? I understand that a black hole's gravity goes beyond the event horizon, but what causes the sudden difference between the event horizon and the area beyond it and why doesn't the event horizon expand to the edge of the gravitational field.
 A: You're probably asking a simple question to understand what is the event horizon, and it appears you think it's maybe where there is no longer an attractive gravitational force from a body. 
Maybe that's not your question or doubt, but either way I try a simple intuitive answer
First, gravity from a body acts to infinity, it just keeps getting weaker and weaker. At some distance far enough it is so weak that you might not be able to feel it's tug, or measure it. The earth's gravity at its surface, denoted by g, is its acceleration of 9.8 meters per second square. But if you go far away it goes down rapidly (it goes down as 1 over the distance to the center of the earth squared). One example will make it clear: gravity from Jupiter at its surface is 2.4 times more than on earth but the force on jupiters gravity on you at the surface of the earth would only contribute 0.0000035% to your weight (pulling you in the direction of Jupiter). So small you can notice it. 
Similarly, the gravitational force or field of a sun-like star goes down to where when you're further out than a distance of a few AUs (astronomical units, the distance from the earth to the sun), you won't notice it as weight (or pulling you up). If you are freely floating, not on earth, yes, it'll pull you like a planet, you'll orbit around it, or fall into it. 
The point is that gravity gets weaker the further out you are from an astronomical body, and far enough away for most purposes (like weighing yourself on a scale) you can ignore it. But it is always there. There is no edge as others have said. 
The horizon is a different kind of beast. It is an imaginary surface around an astronomical body (and only relevant for black holes, that is, for objects that are very very dense and relatively small -- for instance 50 times the mass of the Sun but all compressed inside a diameter of less than 150 Kms. Very very dense. 
It turns out that if that were the case we would not be able to see inside that 150kms diameter surface. Once a star collapses inside its horizon, we just don't see it. The horizon is the surface where no light can escape from inside it. And neither can anything else. Once it falls in, it can't get out, and we can't see it from outside. It's like the horizon of a boat disappearing over the horizon, it disappears. 
Now, this description is a bit simplistic because time starts passing slower as I see the star collapse to its horizon -- it slows down, and I might never see it passing the horizon. But ignore that. The horizon is an imaginary surface beyond which (going inwards), if you were falling in, you'd never be able to get out, nor in fact message anybody outside - if you turned on a very very strong flashlight, the light would never be seen outside the horizon. The light may approach the horizon but never gets past it. 
It happens because the force of gravity is so strong that the escape velocity is higher than the speed of light. And since nothing goes faster, nothing can escape. Now, this is a sort of classical Newtonian view of a black hole, but in reality the strong gravitation bends the geometry of space and time so much that everything inside the horizon simply falls back in, even when it tries to go outwards. 
Outside the horizon gravity is less strong, and someone could come close, not get to the horizon, and go back out (but you will need lots of thrust in your rockets)
So that's the horizon. Hope this simple view gives you an intuitive understanding. The horizon is simply the boundary of no return. Hope this isn't too simplistic, and I didn't presuppose something you didn't mean. This was just meant to be a fairly basic and intuitive explanation.  
See the wiki article on black holes and including something on their event horizons (that's what they are called, there are other kinds of horizons in for instance cosmology) for a richer view and understanding.  https://en.m.wikipedia.org/wiki/Black_hole 
A: The gravitational field is continuous and extends to infinity.  There is no edge to the gravitational field.
The term "event horizon" means the surface surrounding a volume of space from which it is impossible to leave.  Gravitational fields are so high that all possible paths you can take within that volume will lead to it's center.  Once an object passes the event horizon it can never escape that volume of space.
But while that sounds like a specific barrier, if you were the object falling through the black hole (and could somehow survive the extreme conditions) you would not notice any physical surface, because there is none.
So it's not sudden from the point of view of someone passing through it.
It's sudden in then sense that it denotes the place from which you can no longer escape.  In that sense it's like the point on a runway where a plane cannot slow down enough to stop before running out of runway.  There's no physical barrier, but there's a point at which it's no longer possible to do something.
So if I pass through the even horizon it is not, in itself, a sudden event, but when I passed it I suddenly no longer have any way of leaving.
A: To add to Bob Bee's answer and his comment:

"... the strong gravitation bends the geometry of space and time so much that everything inside the horizon simply falls back in, even when it tries to go outwards. "

Specifically, inside the horizon, the future lightcone of every event is contained within the horizon. So everything's future lies within the horizon. You can no more leave the space enclosed by the horizon than travel backwards in time: it's tantamount to the same thing. 
You can visualize the situation with the future lightcones of points along a line leading inwards towards the hole from infinity, crossing the event horizon on the way. To do this precisely, we need to imagine a small volume of spacetime around a freefalling observer at each such point, small enough that the spacetime within it is Minkowskian. 
Far from the hole, most of the geodesics whose tangents are contained in the future lightcone lead to places other than the hole; to such an observer, the horizon subtends only a tiny solid angle. But some of the geodesics - those contained within a nonzero solid angle bundle, will ultimately wind up at a single point in the hole - the singularity (at least in classical General Relativity theory).
Now we move along the line towards the hole and witness what happens to the analogous light cones. As we move along the line, the light cones "tilt" towards the hole as we go so that a greater and greater fraction of the geodesics contained within the light cones wind up ending at the singularity. Finally, we reach a point where all the geodesics lead to the singularity. Even though we are talking only about geodesics, it should be clear that at this point, all possible futures also now wind up at the singularity, for all possible points in our little almost-Minkowskian spacetime region are reachable along a geodesic and general noninertial motion is simply describable as continuously changing geodesics to ride on. This point, where we first reach a condition of all futures' ending on the sigularity, defines a point on the event horizon. If we did this process for all points outside the hole, we'd see that the points defined in this way make up a three dimensional hypersurface - the event horizon - that defines an enclosed volume wherein futures of all points end at the singularity and for all of whose outside points there are still non-singularity-intersecting futures possible. 
The light cones of events exactly on the event horizon are precisely tangent to this special hypersurface so that they are barely contained within the horizon.
There is a rather spiffy hand-drawn diagram of these ideas in Figure 27.11 of Roger Penrose's "Road To Reality" as well as more details in sections 27.8 and 27.9 of the same text.
