Does time slow down because the universe is expanding at an accelerating rate? If the universe is expanding at an accelerating rate, such that the galaxies' moving away from each other is accelerated, then time should also slow down.
And when universe will accelerate to the speed of light, then time should stop.

Does time slow down because universe is expanding at an accelerating rate?
Can any physicist explains if this is true or not?
Yes, time does run slower for far-away objects, as observed from our point of view; this is a prediction of general relativity. And yes, because expansion accelerates, this time dilation slowly, very slowly, becomes more pronounced (this would happen even if the expansion didn't accelerate, but just continued at the same rate).

This time dilation  is a well-known effect, and is always taken into account when doing observations. For instance, when observing distant supernovae, one is often interested in how their luminosities decrease as a function of time. This is called their lightcurve. In order to compare lightcurves at different redshifts, they are usually converted to their restframe, i.e. how they would look if you were "standing next to the supernova" (e.g. Goldhaber et al 2001).

However, time dilation does not work exactly as you seem to think. At galaxy at a redshift of $z$ has its time dilated by a factor of $1+z$, so time runs twice as slow for a galaxy at, say, $z=3$ than for a galaxy at $z=1$. Galaxies with redshifts larger than $z\sim1.5$ recede faster than the speed of light, and time does not at all stop here. Only for $z\rightarrow\infty$, i.e. at the beginning of time at Big Bang, does the time dilation approach infinity.