Learn Hardcore German: Die Erklärung des Lehrers hilft, die Ursache des Fehlers zu verstehen. - The teacher’s explanation helps to understand the cause of the mistake.

In German, possession or attribution is shown by the genitive case. For masculine and neuter singular nouns you use the article des and add -s (or -es) to the noun:

Lehrer → des Lehrers (“of the teacher”)
Fehler → des Fehlers (“of the mistake”)

There are two dies, one before Erklärung and one before Ursache. Why do they look the same, and how can I tell them apart?

German has four cases, but for feminine nouns the definite article is die in both nominative and accusative. Here:

die Erklärung is feminine singular in the nominative (subject of the main clause).
die Ursache is feminine singular in the accusative (direct object of verstehen).
They look identical, so you distinguish them by their function in the sentence (subject vs. object).

Why is there a comma before die Ursache des Fehlers zu verstehen?

That comma sets off an infinitive clause (Infinitivgruppe) introduced by zu. According to modern punctuation rules, you place a comma before a zu‑infinitive group especially when it has its own object or complements. Here it clarifies that hilft and die Ursache … zu verstehen are separate parts.

What does zu do before verstehen, and why does the verb end up at the very end?

In German many verbs combine with a zu-infinitive to express purpose or result. The zu attaches directly to the infinitive verb: zu verstehen. In both subordinate and infinitive clauses German places the verb (or the infinitive) at the end. That’s why verstehen comes last.

Why isn’t there an object like uns after hilft? Isn’t helfen supposed to take a dative?

Normally helfen requires a dative object (e.g. jemandem helfen). However, in general or impersonal statements the beneficiary can be left unspoken or is understood from context. English does the same when you say “It helps to understand…” without naming who. You could add uns for clarity:
“Die Erklärung des Lehrers hilft uns, die Ursache des Fehlers zu verstehen.”

Could I rewrite this with a dass‑clause instead of a zu‑infinitive?

With helfen you almost always use the zu‑infinitive. A dass‑clause after helfen isn’t idiomatic. If you want to use dass, you need to change the main verb or structure, for example:
“Die Erklärung des Lehrers ist so klar, dass wir die Ursache des Fehlers verstehen.”