Learn Turkish: Şu adam daha şişman ve daha fakir.

Both şu and o can mean 'that'. However, şu is usually used for something you can see and are pointing out right now (like saying 'that man right over there'). O is a bit more general or used for things further away or out of sight.

How does daha work to make words like 'fatter' or 'poorer'?

In Turkish, you don't change the ending of the adjective (like adding '-er' in English). Instead, you just put daha (which means 'more') in front of it. So daha şişman literally means 'more fat' (fatter), and daha fakir means 'more poor' (poorer).

Do we have to repeat daha before both adjectives?

It is very common and sounds natural to repeat daha before each adjective for emphasis, just like in this sentence. However, you could also say daha şişman ve fakir, and people would perfectly understand that 'more' applies to both.

Where is the word for 'is' in this sentence?

In Turkish, the 'to be' verb (am/is/are) is often just left out in simple present tense sentences, especially with the third person ('he', 'she', 'it', or 'that man'). The sentence literally reads 'That man more fat and more poor', and the 'is' is naturally understood.