Learn Swedish: Hon går utan att säga hej då. - She leaves without saying goodbye.

Utan does mean 'without', but when it is followed by an action, Swedish requires att (to). While English uses an '-ing' verb ('without saying'), Swedish uses utan att followed by the infinitive verb (säga). You can think of it as literally saying 'without to say'.

I thought går meant 'walks'. Why does it mean 'leaves' here?

Går does mean 'walks', but it is also the standard everyday verb for leaving or departing on foot. It is very similar to how we use 'she is going' in English to mean that she is leaving.

Is hej då always written as two separate words?

Yes, hej då (goodbye) is always written as two words. It literally translates to 'hello then' (hej = hello, då = then) and is the most common way to say goodbye in Swedish.