بعد أن أعود إلى البيت، أكتب الواجب على صفحة نظيفة.

Because the verbs already show the subject. In أعود and أكتب, the prefix أ- tells you the subject is I.

So Arabic does not need أنا here. You could say أنا أكتب for emphasis, contrast, or clarity, but normally it is omitted.

بعد means after, and أن introduces a following verb clause, roughly like after [someone] does... or after [to]... depending on how you think about it.

So:

• بعد = after
• أن أعود = that I return / returning

Together, بعد أن أعود means after I return.

In Modern Standard Arabic, this is a very normal way to introduce a time clause before another action.

Yes. After أن, the imperfect verb is normally in the subjunctive mood.

So in fully vowelled Arabic, this would be:

• أعودُ = I return / I am returning (indicative)
• أعودَ = I return, after أن (subjunctive)

But in normal unvowelled writing, both are written أعود, so you do not see the difference unless short vowels are added.

So the sentence is grammatically behaving as:

بعدَ أنْ أعودَ إلى البيتِ...

even though ordinary text just writes بعد أن أعود إلى البيت.

Because إلى means to. The verb عاد / يعود often takes إلى when expressing returning to a place:

• عاد إلى البيت = he returned to the house/home

So:

• أعود إلى البيت = I return home / I return to the house

English often says simply return home without to, but Arabic commonly uses إلى here.

Literally, البيت means the house, but in many contexts it naturally corresponds to home in English.

So أعود إلى البيت is often best understood as I return home, even though the Arabic word is literally the house.

Arabic often uses a definite noun where English prefers a more idiomatic expression.

In Arabic, the imperfect tense (أكتب) can express:

• a present action,
• a habitual action,
• or a future action, depending on context.

Here, because the sentence begins with بعد أن أعود إلى البيت (after I return home), the next action is understood as something that happens afterward. So أكتب can naturally mean:

• I write
• I will write
• I write (as a routine) after I return home

Arabic often leaves this to context, while English may choose between present and future more explicitly.

الواجب means the homework or the assignment.

It is the direct object of أكتب:

• أكتب = I write
• الواجب = the homework

In fully vowelled Arabic, it would be الواجبَ because it is the object of the verb.

In ordinary writing, that final case ending is usually not written.

على literally means on.

So على صفحة نظيفة means on a clean page. This is a natural way in Arabic to talk about writing something on a page, sheet, or surface.

You can think of it literally:

• أكتب ... على صفحة = I write ... on a page

This is very close to English usage.

Because it describes صفحة, and صفحة is a feminine noun.

In Arabic, adjectives agree with the noun they describe in gender, number, and definiteness.

So:

• صفحة = page (feminine)
• نظيفة = clean (feminine singular)

If the noun were masculine, the adjective would also be masculine.

Because the phrase means a clean page, not the clean page.

In Arabic, if the noun is indefinite, the adjective is also indefinite:

• صفحة نظيفة = a clean page

If the noun is definite, the adjective must also be definite:

• الصفحة النظيفة = the clean page

So the sentence is talking about writing on a clean page, not on some specific already-known page.

A fully vowelled version would be:

بَعْدَ أَنْ أَعُودَ إِلَى الْبَيْتِ، أَكْتُبُ الْوَاجِبَ عَلَى صَفْحَةٍ نَظِيفَةٍ.

A few things to notice:

• بعدَ ends in -a
• أنْ
• أعودَ is subjunctive after أن
• البيتِ is genitive after إلى
• أكتبُ is indicative
• الواجبَ is the direct object
• صفحةٍ نظيفةٍ are genitive after the preposition على

In normal everyday Arabic writing, these short vowels are usually omitted.

Yes. Arabic could also say:

أكتب الواجب على صفحة نظيفة بعد أن أعود إلى البيت.

That means essentially the same thing: I write the homework on a clean page after I return home.

Starting with بعد أن... puts the time setting first, which can sound natural and clear. Putting it at the end is also completely possible.