Learn German: Ich esse Brot mit Wurst. - I eat bread with sausage.

Just like in English, German leaves out the article when talking about foods in a general, uncountable way. You are eating bread and sausage in general, not one specific bread or a whole specific sausage.

Doesn't the preposition mit always require the dative case? Why didn't the word Wurst change?

You are exactly right that mit always forces the dative case! However, German nouns themselves rarely change in the dative. Usually, it is the article (like der, die, or das) that changes to show the case. Since we dropped the article here, Wurst just looks like its normal dictionary form.

Is it possible to say mit einer Wurst here?

Yes, but it changes the meaning slightly. Ich esse Brot mit Wurst means you are eating bread with sausage meat as a topping or side. If you say mit einer Wurst, it sounds like you are eating bread along with one whole, individual sausage.