I mean the physical picture rather than just the mathematical details.
The concept of time reversibility is quite simple in Classical Mechanics. It means that the laws governing the system are deterministic into the future as well as the past. If we know the complete configuration of the system - positions and velocities of all particles - at a given instant, as well as the laws governing its dynamics, then we can predict the configuration at any time into the future and we are able to know what was the configuration at any time in the past. Both action are done without any uncertainty.
Quantum Mechanics is a non deterministic theory, if we know the configuration of the system at one instant we cannot be sure about the configuration at another instant either into the future or to the past. The "knowing the system's configuration" plays a fundamental role here. It means that a measurement has been done and this breaks determinism of the theory. However, we can still discuss time reversibility if we restrict to the time evolution of the system - or rather its wave function - in the intervals free of measurements. This evolution is in most cases time reversible in the sense that if we know the wave function at a given instant (and the dynamics governing it) we also know the wave function at times in the future and in the past.
A common misunderstanding about time reversal symmetry is to believe that at any given instant the system may spontaneously revert its direction. It will not do this thanks to the existence of conservation laws. A spinning top will never spontaneously invert its spin, even though Classical Mechanics is time reversible. The angular momentum conservation prevents that to happens. Time reversal symmetry in this case just means that if we know the angular velocity now we know the angular displacement for any instant ahead or any instant before.
When we say that time reversal symmetry is broken for systems containing a large number of particles it does not mean that microscopic laws are no longer time reversible. They indeed are. It means that the macroscopic behavior or evolution of the system has a preferred direction to follow. There is no clear threshold for this macroscopic time reversal symmetry breaking given by the number of particles. Moreover, this breaking has a statistical character.
To illustrate this, consider an imaginary plane dividing a box into the left (L) and right (R) regions. If we have two particles traveling around in the box and subject to the laws of classical mechanics, we can easily find the particles (1,2) as (L,L), (R,R), (L,R) and (R,L). These are the microstates of the system. We can define the macrostates as (L), (R) and (M) which means, two particles in the left, two particles in the right and one particle each side. These macroscopic states are the ones cared by Thermodynamics and constrained by the Second Law. All microstates are equally probable but since there are two microstates corresponding to the macrostate (M) it is more probable that we observe the particles one each side of the box. Anyhow, it is pretty easy to find all the particles at one side. It means that spreading out from one side or congregating to it are "easily possible" (in a probabilistic sense) and therefore we say the system is reversible. However as we increase the number $N$ of particles, the number of microstates grows exponentially, $2^N$, whereas the number of microstates corresponding to the macrostates (L) or (R) does not change, it is one. Thus the probability of finding all particles to the left is $1/2^N$ - the same as obtaining $N$ heads after tossing a coin $N$ times. For a gas, with typically $10^{23}$ particles, this probability is nearly zero. The time we would have to wait in order to see all particles at one side would be possibly longer than the age of the Universe. In this sense, this system is not reversible - you will never see all the air in the room spontaneously going to the left side.