A closer look
Yes, we'll is a real word. It is a contraction, a squeezed-together form, of we will (and, in some uses, we shall).
We'll eat at six means we will eat at six.
We'll meet you outside the coffee shop. Say it at the door when a friend asks where you are. It simply means we will meet you there.
We'll eat at six, so come hungry. This is how people talk about plans. In writing, the longer form we will eat at six also works, but the contraction sounds more natural in speech.
One nearby point: in formal writing, such as a job application or an academic essay, writers usually prefer the full form we will, so that is what you would write there. But we'll is still a real word in those settings too, not something only suited to casual contexts.