A butcher needs to have two separate tubs of water – one where he washes the meat and the other where he washes the forbidden fats.
Similarly, a butcher needs to keep three separate knives – one for slaughter, one for cutting meat and one for cutting off forbidden fats.
The reason that the butcher needs a separate knife for slaughter seems clear. Since we do not want the sheḥita knife to develop nicks or other deficiencies that would render it unfit for slaughter, we do not want the butcher cutting meat that might have bones that may ruin the blade. Rashi explains that although the butcher should really check the knife before using it for sheḥita so he would, theoretically, discover any problems with the knife, nevertheless the Sages feared that if he is in a hurry he may neglect to do so. Others suggest that the concern is about the blood that may have been absorbed in the knife at the time of slaughter. If such a knife is used for cutting meat, the meat may absorb that blood and become forbidden.
The Gemara wonders why two separate knives are needed for cutting meat and cutting forbidden fats. Wouldn’t it be sufficient to have a single knife and require the butcher to cut the forbidden fats only after he finished cutting the meat? The Gemara explains that we fear that he may switch the order and cut the forbidden fats first. Once there is a requirement to keep separate knives the butcher becomes sensitive to the need to keep the meat separate from the forbidden fats. The Gemara offers the same explanation for the two separate tubs. Even though the Sages could have simply required that the meat be washed before the forbidden fats, by requiring two separate tubs we raise awareness that they must be washed separately and we are no longer concerned that the butcher will make a mistake.