How does General Relativity explain escape velocities? In general relativity, objects follow the shortest possible path through curved space-time called a geodesic and that there exists no such force of gravity which pulls objects, it is just because their timelines in the warped space-time fabric are towards the object causing the curvature. If this is so, then how are objects able to escape a body's gravitational pull by reaching a certain velocity what we call the "Escape Velocity", do they change their geodesic trajectories through space-time?
 A: Yes. Remember that geoodesics are space-time paths: the velocity is embedded in how they are slanted relative to the time direction. 
So if you look at some point in spacetime, there are geodesics leaving it in all future time-like directions that corresponds to paths a particle emitted in that direction and velocity would take. Some of them may escape the vicinity of a heavy object and go off to infinity. The slowest such geodesic defines the escape velocity.
(Geodesics are often described as the shortest path between a starting and ending point, but you can also define them by a starting point and initial direction and then continue them as long as it is feasible using the geodesic equation.)
A: I read in your profile that you hate mathematics but love it in
physics. It's rather hard for me to understand it, but I don't want to insist on this point. Just some ideas deriving from what I've seen in my long years of practice with theoretical physics.
Just to give you an example. When you talk of geodesics, that's a
genuine mathematical concept. It's wonderful that it found so a
beautiful application in GR. Maybe you don't know that initially
Einstein was reluctant to approach his physical problem (to reconcile gravitation with relativity) using such abstract instruments. So I can hope that continuing your studies you'll change your mind. It's simply impossible to deepen one's knowledge of the topics you like without a serious knowledge of mathematics.
Incidentally you wrote

objects follow the shortest possible path

but the contrary is true. In spacetime a geodesic representing a body's path isn't the shortest path between two assigned points, but the longest. Anders Sandberg very aptly remarked that there is another approach to the concept of a geodesic. The one based on its (maximal or minimal) length among all paths connecting two given points is one, usually referred to as variational approach. 
The other, more useful for your question, characterizes geodesics as
special curves among all those starting from a given point with a
given direction. It's difficult to give you a more precise idea
(without using the relevant mathematics). I'll try the following.
In a flat space or spacetime there are straight lines, with the
properties you know. In a curved space (or spacetime) straight lines
don't exist (geometry isn't euclidean). We can try what in a sense
is the best possible approximation to a straight line, i.e. a curve
that saves some of the properties of the straight line. That it were
possible was discovered few years before Einstein confronted himself
with GR problem. Unfortunately I'm afraid it's impossible to go
deeper. I limit myself to give you two magic words: "parallel transport".
A: Escape velocity is the rate at which space-time is falling into the surface of the earth. The fabric of space-time is literally being absorbed by the earths mass. According to General relativity gravity is not a force. Rather objects follow a straight line and when they are near a massive object the space they are in gets bent towards the matter as it moves forward in time. We who are standing on a rigid earth feel what seems like gravity but is actually only our acceleration against the inflow of space-time. Escape velocity is the additional motion required to overcome the accelerating inflow of space-time.
