The balloon analogy thread seems a good place to start. But I could attempt sketches to some of this as best I understand it:
aditya ver.2.0 said:
My question is that what is the source of this mysteries force? People stay that it existed ever in the universe.
Dark energy is believed to be the energy of the vacuum. And our inflationary cosmology has a vacuum as far back as we know about.
aditya ver.2.0 said:
So how did the cosmos crunch into a volume of nutshell before the Big Bang?
It didn't have to, it has also been expanding as far back as we know about.
Are you confusing the observable universe with the (local) universe? The former had to have a volume smaller than a proton when we go as far back as we know about. The latter is at least 1000 times larger but we don't know how large it is. It could, arguably, be infinite.
aditya ver.2.0 said:
And how is constantly increasing its energy and increasing the rate of expansion much further?
Why would the universe increase in energy? The universe has no measure of global energy that enough people can agree on. (And when they agree on something such, it turns out it doesn't change due to that general relativity balances the terms AFAIU.)
If you think the mystery is where various energy "comes from", you have it easy. It doesn't seem to be the problem, if the universe expands it will continue to do so. The real mystery is where spacetime "comes from" as it expands. Who ordered that, that it can expand!?

Yet it has to be able to do so, else we can't have the same physics everywhere, this is what general relativity guarantees us.
The increase in expansion is simply (well...) what happens when vacuum energy dominates according to general relativity. When matter and/or radiation dominates the expansion rate changes differently. It is internal phase changes that decides the behavior, akin to how a gas or a liquid behaves differently as you push a piston in a cylinder. In fact, an isolated ICE cylinder is the best analogy I can think of - you have dumped in fuel and oxidizer and can burn/not burn so different expansion behavior - with no energy exchange with the "outside".