Learn Portuguese (Portugal): Que dias é que trabalhas?

In Portuguese, when asking "what" or "which" immediately before a noun like dias, you use que. You use quais when asking "which ones" out of a specific group. So "which days" translates directly to que dias.

What does "é que" do in this sentence?

The phrase é que literally translates to "is that", but here it acts as a common question tag. It does not translate into English. Portuguese speakers use é que after question words to make the sentence sound natural and to keep the normal word order.

Shouldn't there be a word for "on" like "on which days"?

In English you say "on which days", but in Portuguese you can often ask que dias directly without a preposition. You might hear em que dias sometimes, but leaving out the em is completely natural and very common in everyday speech.

How do we know it means "do you work" if the word for "you" is missing?

The verb trabalhas ends in -as, which is the specific ending for tu, the informal word for "you". Because the verb ending already shows who is working, the word tu is dropped, which is standard practice in Portuguese.