Learn Hardcore Turkish: Literatürü inceledikçe konuyu daha iyi anlıyorum. - The more I examine the literature, the better I understand the topic.

The suffix -dikçe is a temporal/concessive converb that attaches to a verb to express “as/when I do X” or “the more I do X, the more Y.” Here, incele- (to examine) + -dikçe → inceledikçe, meaning “as I examine” or “the more I examine.”

How exactly is inceledikçe formed, and why is it -dikçe instead of -dıkça?

Start with the verb stem incele-. Then add -dikçe. Because incele- ends in a front vowel e, Turkish vowel harmony requires the front form -dikçe (not -dıkça). So you get:
incele- + ‑dikçe → inceledikçe

Why is Literatürü in the accusative case with -ü?

In Turkish, a definite/specific direct object takes the accusative suffix -(y)I (with four-way vowel harmony: -ı, -i, -u, -ü). Literatür ends in ü (a front rounded vowel), so it takes -ü:
literatür + ü → literatürü
This marks “the literature” as a specific object.

Why does konuyu also carry the accusative -yu?

Konuyu (“the topic”) is the direct object of anlıyorum (“I understand”). Since it’s definite/specific, it receives the accusative suffix -(y)I. The final vowel u in konu triggers -yu:
konu + yu → konuyu

What does daha iyi mean here, and why can’t we just say iyi?

İyi means “good.” Daha iyi literally means “more good,” i.e. “better.” Because the sentence expresses a comparison (“the more I examine, the better I understand”), you need the comparative daha.

What tense/aspect is anlıyorum, and why is it chosen?

Anlıyorum is the present progressive/simple present, formed with -(I)yor + -um. Turkish uses this form both for ongoing actions (“I am understanding”) and habitual/general truths (“I understand”). Here it conveys a repeated or developing result: “I (now) understand (better).”

Could I change the word order, for example putting konuyu first?

Yes, Turkish word order is flexible. The standard is Subject-Object-Verb, but adverbial clauses often come first. The neutral phrasing is:
Literatürü inceledikçe konuyu daha iyi anlıyorum.
If you front konuyu for emphasis, you could say:
Konuyu literatürü inceledikçe daha iyi anlıyorum.
The meaning stays the same, though the focus shifts slightly.

Why is there no explicit subject pronoun like ben?

Turkish is a pro-drop language: the verb ending -um in anlıyorum already indicates “I.” Adding ben is unnecessary unless you want to emphasize the subject:
Ben literatürü inceledikçe konuyu daha iyi anlıyorum. (It’s me who understands …)