In the sentence, "Prepara tú el café, yo los voy sirviendo" why is the tú after the verb?
If you mean why it appears, the reason is that you can always include them. Omitting subjects is possible, but not compulsory. In this case, adding the pronoun highlights that it is you to whom the sentence is directed. It is an order. The speaker is "setting the task distribution", and this is very likely to have such interpelations.
You, prepare the coffee.
If you meant why it is after the verb, it's because in Spanish you're free to put it before or after the verb in questions (and in general, but try to speak normally ;) )
Prepara works as an imperative form. It is a order for whoever.
However, the pronoun
tú, is not required because it is implicit in the verb. The form
prepara it's a 2nd form imperative:
The sentence has exactly the same meaning if you exclude
tú, it is just a way of speaking. Nevertheless, it is common to use pronouns to reinforce imperations.
The pronoun is optional and movable; it can appear before or after the verb, as in most cases in Spanish. The exact meaning of the phrase is subtly different according to whether you use the pronoun or not, and if you do use it, where you put it.
If the pronoun comes before the verb, at the beginning of the sentence, it can be interpreted as a vocative or a topic. That is, if you say tú prepara el café, then you can be interpreted as calling or addressing the other person ("hey, you, prepare the coffee"), or as making it the new theme of the sentence ("as for you, prepare the coffee"). In the first case a good practice is to write it with a comma after the pronoun (tú, prepara el café), although this is increasingly not being done in informal communications.
If the pronoun comes after the verb, the meaning tends to be the usual meaning of pronouns that are not dropped when they could be, i. e. contrastive or emphatic: prepara tú el café means "you prepare the coffee [because nobody else will]". In your example the contrast is made clear by the following clause: prepara tú el café, yo los voy sirviendo commands a division of tasks, "you do this, I will do that".
This is all based on the Nueva Gramática de la Lengua Española, §42.4b-c:
As a supplement to the existing answers:
Just as in English, adding the optional pronoun to the imperative provides emphasis, which in this case is desirable because you're comparing what each person is going to do.
(1) Can the pronoun be omitted? Yes, but. Let's look at this.
Prepara el café; yo los voy sirviendo. | Make the coffee; I'll do the serving.
It's okay but your sentence makes the teamwork and division of labor clearer.
(2) Can the pronoun go before the verb? Yes, but then you need an audible comma.
Tú, prepara el café; yo los voy sirviendo. | You, make the coffee; I'll do the serving.
Sounds a bit rude in either language. If you want to do it this way, it would be more polite to use the given name instead of the pronoun, and to add a "please":
Víctor, prepara el café, por favor; yo los voy sirviendo. | Victor, make the coffee, please; I'll do the serving.
But the original sentence as you gave it is perfectly polite as is.