How exactly does the curvature of space-time cause objects to "fall" into another object Let me preface this by saying that I'm a high school student so my knowledge of GR might be seriously flawed.
As far as I'm concerned, Einstein's theory of GR states that matter curves the fabric of space-time so instead of there being an imaginary "tug," objects are simply falling in.
My question is, how do the objects "fall" in because in real-life simulations of the curvature of space-time, the force that's causing objects to "fall" in is Earth's gravity. In other parts of space, what causes objects to "fall" in. Does the Newtonian "tug" of gravity still exist within Einstein's theory because, without it, I can't seem to understand why curving space-time would cause stuff to "fall" in?
 A: The most important of General Relativity, which is generally not told by the popular science educators (where I am assuming you heard about General Relativity), is that objects always follow the shortest path through space-time (remember that objects are always moving through space-time). When there is no energy or matter in the surrounding space-time then objects follow straight lines since space-time is flat. Now if you have some matter or energy to curve space-time then the shortest path isn't a straight line. The shortest path between any two points on a curved space is called a geodesic. All the falling body is doing is following the geodesic.  
A: Don't think of "falling in". Instead, think of "moving in a straight line". I assume you've learned Newton's la of inertia, so it's clear and intuitive to you that if a puck moves on a frictionless horizontal plane it will just carry on moving on a straight line at constant velocity.
Now to think of GR, people tell you to think about a curved surface, rather than a plane. But the important thing to realise is that this is just an analogy. The real "curvature" isn't like that, but is abstract. 
What you should do, is imagine this situation from above. In the Newtonian case, the puck just continues in a straight line. In the GR case, the puck appears to move in "non-straight" line. Ignore the fact that in the GR the surface is "curved" - it isn't really bulging in real-space, that's just an analogy to make the trajectory from above look curved. The point of it is that now the puck moves in a "line" that appears curved to us. No forces are acting, the puck is still moving inertially along the line of inertial movement. 
One way to say it is to treat the inertial-movement line as the definition of a straight line. So you keep the Newtonian idea that inertial bodies move along a "straight line", only now this line appears curved in 3d-space. This is like how a circumfrance-line of a sphere is a "straight line" along it, i.e. it's the shortest way to connect two points along it, even though the line is actually curved in 3d-space.
A: You might want to be inside a windowless falling capsule , for the context of a GR discussion. Maybe you are in orbit around the earth, or actually falling down.  You would almost not feel gravity. However, you would see the tidal effects on a puff of smoke that would change shape over time. The smoke particles are following geodesics. Analysis of these geodesics will reveal the presence of earth and possibly its motion. You may want to set the origin of your space-time coordinates in the center of your capsule now. You may want to open a window to see if the earth is rotating below you or moving toward you to give you a good smack on the future branch of your time axis.
As for throwing a rock out the window, it will follow a space-time geodesic, that you can express in the coordinate system centered in your capsule. If you draw a graphic of its space-time trajectory, you will need extreme scaling on the time axis. Light goes really fast. 
Instead of answering your question, I have asked you to change your context. Think of tidal forces rather than throwing rocks.  
