How can an inflaton field let spacetime expand? Whatever the Nature of an inflaton-field, how did it make spacetime grow at an incredible rate? A little speck of spacetime was inflated to the entire observable Universe and everything beyond in very little timespan, starting at $t=10^{-36}$ and ending at $t=10^{-35}$ (correct me if I'm wrong). 
How did spacetime and the inflaton-field connect? Was it a kind of vacuum energy which makes spacetime nowadays expand at an increasing rate (according to the observations of supernovae, though I strongly doubt that this forms a proof of an accelerated expansion). 
 A: An inflaton field is usually assumed to be an almost-spatially-homogeneous scalar field $\phi(t)$ with a self-interaction $V(\phi)$. Such a field can be shown to have energy density
$$\rho=\frac{1}{2}\dot{\phi}^2+V(\phi)$$
and pressure
$$P=\frac{1}{2}\dot{\phi}^2-V(\phi).$$
When the field is slowly-varying in time, the approximate equation of state is thus
$$P=-\rho.$$
This is very different from the equation of state for non-relativistic matter, $P=0$, and the equation of state for radiation (and ultra-relativistic matter), $P=\rho/3$. In fact, it is the same equation of state as for dark energy.
The negative pressure is what causes the exponential expansion when you solve the Friedmann equations. It isn’t that it “lets spacetime expand”; it makes spacetime expand.
The exponential expansion during inflation was extremely rapid because the interaction energy of the inflaton field is theorized to be very high: perhaps $10^{16}$ GeV, only a few orders of magnitude below the Planck energy.
The exponential expansion of the dark energy era is extremely slow because dark energy apparently has an abnormally low energy density which physicists are struggling to understand.
