Learn Spanish (Spain): Dijo que aquel día estaba muy cansada.

We know it's a 'she' because of the adjective cansada at the end of the sentence. Since it ends in -a, it tells us the person who was tired is female. Assuming she is talking about herself, dijo must mean 'she said'.

Why use aquel instead of ese to say 'that'?

Both ese and aquel mean 'that', but aquel refers to something further away in time or space. When talking about the past, Spanish speakers often use aquel día (that day) to emphasize that the day feels distant from the present moment.

Why is 'was' translated using the imperfect tense (estaba) rather than the preterite (estuvo)?

Being tired is a temporary physical state or condition. In Spanish, we use the imperfect tense (estaba) to describe background states or how someone was feeling in the past, because there is no focus on a specific start or end to the tiredness.

In English, we can say 'She said she was very tired.' Can I drop the word que in Spanish?

No, you can never drop que when it connects two parts of a sentence like this. While English frequently drops the word 'that', Spanish strictly requires que to link the first action (Dijo) with the rest of the sentence (estaba muy cansada).