How did inflation lead to the expansion of space? Big bang Theory gives a scientific explanation of the evolution of the Universe, where the scientific picture of it starts with Universe(space) uniformly filled with matter and energy, already expanding without any description of how that expansion started. 
Alan Guth in this course states inflation as a Bang which Big bang theory doesn't give any description about. 
According the theory of cosmic inflation Universe undergone exponential expansion which was due to repulsive gravity between the patch of gravitationally repulsive unstable material which decayed like a radioactive substance ending the exponential expansion to get our local Universe to the state where the Big bang theory gives the solution from (where the space is already expanding).
General relativity explains the Gravity as the geometry of Spacetime. If repulsive gravity is thought to be working the same way then how did gravity make the space continue expanding (as we mean by metric expansion) and how the space is expanding even today or If I am wrong how repulsive gravity work as the theory of inflation explains. I'm asking this as I previously mentioned Alan Guth called the inflation as Bang to the Big Bang or the event that drove the expansion of space. 
 A: 
Big bang Theory gives a scientific explanation of the evolution of the Universe, where the scientific picture of it starts with Universe(space) uniformly filled with matter and energy, already expanding without any description of how that expansion started.

Big Bang is a theoretical model within the theory of General Relativity. It was developed to explain the observation that all the clusters of galaxies are receding from each other and from us. This behavior is describable by an explosion in four dimensional space.


Extrapolation of the expansion of the universe backwards in time using general relativity yields an infinite density and temperature at a finite time in the past.[18] This singularity signals the breakdown of general relativity and thus, all the laws of physics.


Forty years ago when I took a semester of cosmology that was the dominant model. Since then new observations showed that the model was inconsistent with the Cosmic Microwave Background observations, which showed a surprising uniformity of the matter distribution at the time of the decoupling of the photons, at 380.000 years from the beginning of the Universe. The uniformity could not be fitted thermodynamically because at that age not all parts of the universe could interact with others, due to the light cones of special relativity.


Alan Guth in this course states inflation as a Bang which Big bang theory doesn't give any description about. According the theory of cosmic inflation Universe undergone exponential expansion which was due to repulsive gravity between the patch of gravitationally repulsive unstable material which decayed like a radioactive substance ending the exponential expansion to get our local Universe to the state where the Big bang theory gives the solution from (where the space is already expanding).

The model now , at times smaller than 10^-32 seconds has incorporated an effective quantization of gravity (quantization of gravity is an ongoing theoretical research project). This introduces randomization of the available energy density by a particle called "inflaton" ,    and  small inhomogeneities also appear due to the intrinsic probabilistic nature of quantum mechanics; this  develops into the CMB spectrum and the matter distribution we observe at present.
In the figure, the original Big Bang solution would start after the inflation period. Measurements since 1990 show that the universe is continuing in accelerated expansion, (seen in the figure as an opening horn shape) so the Big Bang name is just a historical terminology in the development of models for the universe, which is retained as  a historical relic.
Expansion, the inflation periods, can be modeled with "negative gravity" mathematically, as gravity is attractive and leads to contractions.

General relativity explains the Gravity as the geometry of Spacetime. If repulsive gravity can be thought the same way the how did gravity make the space continue expanding (as we mean by metric expansion) and how the space is expanding even today or If I am wrong how repulsive gravity work as the theory of inflation explains.

There are two models: the underlying general relativity  one of gravity as distortion of space time, and the other is to view the emergent effects as Newtonian gravity, since locally one can always approximate with Newtonian mechanics the geometry of general relativity.  Gravity effects induce accelerations towards contraction. The inflation period is accelerated expansion. Thus since normal gravity is attractive, negative  gravity is  repulsive: the effect of accelerating expansion translated to a Newtonian mathematical  framework can be modeled  as negative gravity .
The observed continuing acceleration of the expansion of the universe, as seen in the figure, has introduced dark energy in the mathematical models, which has in newtonian mathematics a negative gravity effect.


In physical cosmology and astronomy, dark energy is an unknown form of energy which is hypothesized to permeate all of space, tending to accelerate the expansion of the universe. Dark energy is the most accepted hypothesis to explain the observations since the 1990s indicating that the universe is expanding at an accelerating rate

I'm asking this as I previously mentioned Alan Guth called the inflation as Bang to the Big Bang or the event that drove the expansion of space.

Yes, as you can see in the attached figure of the time development.
