Why do we need inflation? wiki (https://en.wikipedia.org/wiki/Inflation_(cosmology)) says that immediately after BB there was expansion at speed greater than $c$, what makes this necessary, what would happen if expansion took place at near c?
 A: There are three commonly-given reasons to believe in cosmic inflation, and they're all given in the Wikipedia article.

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*The horizon problem. The universe is homogeneous on large scales when there is no reason it should be.

*The flatness problem. The universe is surprisingly close to flat when there is no reason it should be.

*The monopole problem. Magnetic monopoles are predicted to be copious, but we've not found any. If the theories are right, then they've presumably been diluted (by inflation).

See the Wikipedia article for more details.
A: Cosmic inflation is an expansion of the metric of space itself, not an expansion or explosion into space like a supernova. As such, the rate of expansion is not limited by the speed of light.
The evidence for the expansion of space is the redshift of light from distant galaxies, and in particular the relation between redshift and distance known as Hubble's law. The fact that redshift only depends on distance an not on direction shows that either space itself is expanding or we are in a unique location from which all galaxies are receding (which is extremely unlikely).
A rapid exponential expansion of space in the first $10^{-32}$ seconds of the universe is required in order to reproduce the observed smoothness of the cosmic microwave background; the finely tuned density of matter and energyin the universe (the flatness problem); and the failure to find evidence of magnetic monopoles in the observable universe.
