Chullin, Chapter 8, Mishnah 2. This chapter is devoted to the prohibition of basar bechalav, meat cooked with milk, and to the safeguards the chachamim built around it. The previous mishnah dealt with meat and cheese sharing one table; this mishnah continues in the same direction, first with the two foods sharing one cloth, and then with two diners sharing one table.
The opening ruling reads: "Tzorer adam basar ugvinah bemitpachat achat" - one may tie up meat together with cheese inside a single cloth. And then comes the qualification: "uvilvad shelo yiheyu nog'in zeh bazeh" - provided only that neither one makes contact with the other.
So the wrapping itself is not the problem. Meat and cheese may travel together in one napkin. What is forbidden is any contact at all between them, and this is true even when both are cold, where there is no question of cooking one into the other.
Why should cold contact matter? Because at the point where the meat pressed against the cheese, a trace of each may well have rubbed off onto the other. The surface of the meat now carries a bit of cheese, and the surface of the cheese carries a bit of meat. Whoever eats them will be swallowing the two together, and eating meat and cheese together in this fashion is an issur derabanan, a rabbinic prohibition.
The remedy is simple enough: rinse off the spot where the two touched, and the food may be eaten. But the chachamim were not prepared to rely on that. A person may easily forget to rinse, take a bite, and find himself transgressing a rabbinic prohibition by consuming the two of them at once. To keep that from happening, the mishnah rules out contact entirely.
The mishnah continues with a second ruling in the name of Rabban Shimon ben Gamliel: "Shnei achsana'in ochlin al shulchan echad" - two guests lodging at an inn may take their meals at one shared table, "zeh basar vezeh gvinah, ve'ein choshshin" - the one with meat set before him and the other with cheese, and we raise no objection.
Imagine two wayfarers put up at the same inn, complete strangers to one another. Nothing stops them from sitting down together, the first served a meat meal and the second served cheese. There is no fear that the two foods will end up joined, nor any fear that either man will reach across, take what belongs to the other, and combine it with what is on his own plate. People who have never met do not eat from each other's portions.
That is the point of contrast with the previous mishnah. There, the concern was that people eating at one table draw from one another's dishes, so that meat and cheese set out together are likely to meet. Here, where the two diners are unacquainted, that entire chashash falls away, and the halachah follows accordingly.