What they mean is that you will be able to return the bodies to their original states, but not without causing a change in the physical state of something else, like, for example, the states of one or more ideal constant temperature reservoirs. As an example, if you put the body that was originally higher in temperature into contact with a reservoir at its original higher temperature and the body that was originally lower in temperature into contact with a reservoir at its original lower temperature, then, in the end, both bodies will be at their original temperatures. However, the states of the two reservoirs will have changed. Even if you use more than two reservoirs (say at intermediate temperature), in the end the states of the various reservoirs will have changed.
If, instead of initially putting the two bodies into direct contact with one another to let them equilibrate, you kept them separate, and brought each of them into contact with a sequence of reservoirs (different for the high temperature body from the low temperature body) running gradually from their initial temperatures to their final temperatures (at the final equilibrium temperature they would have reached by allowing them to irreversibly equilibrate), this would have been a reversible process. You could have reversed this change and gotten both bodies back to their original states, as well as all the reservoirs.
ADDENDUM
Suppose the cold body started out at 0 C and its final temperature was 100 C. Now we wish to consider several alternate processes for raising the temperature of the cold body from 0 C to 100 C, and we want to see how close we can come to doing this reversibly (so that it can be returned to its initial state without significantly changing anything else).
The mass of the body is M and its heat capacity is C. We have at our disposal an array of 101 ideal constant temperature reservoirs, running in equal increments of 1 C, from 0 C for reservoir #0, to 100 C for reservoir #100.
PROCESS 1:
Forward path: Put body into contact with reservoir #100 and let them equilibrate. Heat transferred to body from reservoir 100 = 100MC
Reverse path:
Put body into contact with reservoir #0 and let them equilibrate. Heat transferred to body from reservoir 0 = -100MC
The net result of this process for the forward and reverse paths is a transfer of 100MC heat from reservoir #100 to reservoir #0. So the process is far from being reversible.
PROCESS 2:
Forward path: Put body into contact with reservoir #50 and let them equilibrate. Heat transferred to body from reservoir #50 = 50MC. Then put body into contact with reservoir #100 and let them equilibrate. Heat transferred to body from reservoir #100 = 50MC.
Reverse path:
Put body into contact with reservoir #50 and let them equilibrate. Heat transferred to body from reservoir #50 = -50MC. Then put body into contact with reservoir #0 and let them equilibrate. Heat transferred to body from reservoir #0 = -50MC.
The net result of this process for the forward and reverse paths is a transfer of 50MC heat from reservoir #100 to reservoir #0 (reservoir 50 remains unchanged). So the process is still pretty irreversible, but less so than Process 1, which results in a transfer of 100MC from reservoir #100 to reservoir #0.
Process 3:
Forward path: Put body into contact with reservoir #1 and let them equilibrate. Heat transferred to body from reservoir #1 = MC. Then put body into contact with reservoir #2 and let them equilibrate. Heat transferred to body from reservoir #2 = 1MC. Continue this through the sequence of reservoirs in 1 C increments until the body reaches 100 C.
Reverse path:
Put body into contact with reservoir #99 and let them equilibrate. Heat transferred to body from reservoir #99 = -1MC. Then put body into contact with reservoir #98 and let them equilibrate. Heat transferred to body from reservoir #98 = -1MC. Continue this through the sequence of reservoirs in 1 C increments until the body reaches 0 C.
The net result of this process for the forward and reverse paths is a transfer of 1MC heat from reservoir #100 to reservoir #0 (all reservoirs in between remain unchanged). So this process is not nearly as irreversible as the other two processes which involved much larger temperature changes with many fewer reservoirs.
Now imagine extending this game plan to an infinite number of reservoirs in vanishingly small temperature changes. In this limit, the process becomes fully reversible.