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It is not prohibited, but, traditionally, some people have chosen not to eat meat in the hope that it makes the heart softer towards the tragedy of Imam Husain (A), or as a gesture of abstinence for mourning, or because Imam Husain (A) was slaughtered similar to how an animals is slaughtered, or some other reasons. 

Anyway I am not aware of the origin of this practice, but it is a personal choice whether one eats meat or does not eat meat.

In practice, many of the traditional foods distributed in some places during the month of Muharram (such as the stews made in big pots) contain meat.