Learn Hardcore Turkish: Zaman kazanayım diye taslağı dün akşam hazırladım. - I prepared the draft last night so I would save time.

Hazırladım: plain completed past, speaker-witnessed.
Hazırlamışım: reported/experiential past (“apparently I prepared it” or you realized later).
Hazırlamıştım: pluperfect (“I had prepared it,” earlier than another past reference). Here, plain past is the natural choice.

It’s the 1st person singular optative/volitional: -A(y)ım = “let me / that I may.” So kazan-ayım means “that I (can) save/gain.” After diye, it expresses purpose: “so that I may save time.”

What does diye mean in this sentence?

Here diye introduces a purpose clause and is best read as “so that / in order that.” The clause before diye is finite and marked for person: Zaman kazanayım diye … = “So that I (may) save time, …”

Can I say zaman kazanmak için instead of zaman kazanayım diye?

Yes: Zaman kazanmak için taslağı dün akşam hazırladım is fully correct. Nuance:

… kazanayım diye feels a bit more immediate/volitional and shows person directly in the purpose.
… kazanmak için is a neutral, more nominal style (with the infinitive). You can also mark other subjects with both: Onlar zaman kazansın diye … / Onların zaman kazanması için …

Why is it taslağı (accusative) and not just taslak?

Because the direct object is definite/specific. In Turkish, definite direct objects take the accusative:

Taslağı hazırladım = I prepared the draft (a specific one).
Taslak hazırladım = I prepared a draft (non-specific/indefinite).