I'm reading through Kardar's Statistical Mechanics of Particles; in the section 1.5 he says:
A reversible process is one that can be run backward in time by simply reversing its inputs and outputs. It is the thermodynamic equivalent of frictionless motion in mechanics. Since time reversibility implies equilibrium, a reversible transformation must be quasi-static, but the reverse is not necessarily true.
Why time reversibility implies equilibrium ?
Consider equilibrium from the point of view of kinetics. Many "moving parts" in the system combine to give the picture of a macroscopic system in equilibrium.
In such systems (dynamic) equilibrium is defined as the situation in which forward and backward processes proceed at equal rates -- thus the macroscopic system doesn't change with time. So time reversal symmetry (being able to swap the forward and backward direction since they both have equal rates) would imply that the system is in an equilibrium state.
Maybe, an author of your book assumed, that possibility of transition of free macroscopic system from equilibrium state to non-equilibrium is small, because non-equilibrium states number is much less then equilibrium states number for free system. So probability of finding system in the equilibrium state is increasing with time. Then it means that if your system isn't in equilibrium at the same moment of time, the possibility of transition to previous moments is small, and so "time reverse possibility" is also small.
So if you want to reverse the time for your macroscopic system, it must be in the equilibrium for the long period of time.
Also I just copy my comment, because it is more formal answer on your question.
It refers to the second law of thermodynamics. According to it, entropy of the closed system isn't decrease. So if you look at an irreversible macroscopic process, where $\delta S > 0$, you can't reverse time, because it leads to entropy decreasing, which contradicts the second law (by the other words, thermodynamics laws aren't symmetric under time inverse operation). But when the system is in equilibrium, it means that macroscopic thermodynamical values don't change, so $\delta S=0$, and theoretically you can reverse the time.
Time reversibility doesn't imply equilibrium per se. The second law of thermodynamics isn't an absolute physical law. It is a probabilistic law, meaning that the balance of probability favors increasing entropy over decreasing entropy, simple because the number of unordered states for any system is far greater than the number of ordered states. If you start with a stack of cards numbered 1 through 100, arranged in ascending order, and shuffle them, the arrangement will fall apart, and most likely it will keep falling apart. Yes, theoretically you could start with a random deck and shuffle it and it would be sorted, but that's like one in a trillion case. (Actually the math suggests that it would be a one in a 100! [that's a factorial] cases, which would be approximately equal to 1 followed by 158 zeroes, but this is a physics discussion not math) So in all practical use cases, the second law of thermodynamics holds.
Now for a process to be time reversible the change in the Gibbs free energy must be zero. This is because $\Delta$G represents the driving force behind any process. If this is negative, meaning the free energy is decreasing, the process is spontaneous. If this is positive, for the process to occur there must be external influence on the system. In the borderline case, there is no need for an external influence on the system but the process doesn't want to procceed in any direction i.e it isn't biased. Each component, for example individual molecules if we are talking about a chemical reaction, proceeds in either the forward or reverse direction at random. Now it is still possible for the process to proceed in any one direction, just as it is possible to get a trillion trillion consecutive heads on just as many tosses, we can assume it won't happen. By the way, trillion trillion isn't that big a number, in the context of molecules, it is $10$$^2$$^4$ or about one and a half moles. The equilibrium thus achieved is a result of half of the tosses being heads, and the other half being tails. Imagine losing one buck per head and gaining one buck per tail, you almost come out even. This equilibrium is known as dynamic equilibrium because there actually is exchange of money happening, you just don't see it on the macroscopic scale.
