הוא כועס עליי, אבל אני לא יודע למה.

כועס literally behaves like an adjective/participle meaning angry. In modern Hebrew, words like this are often used where English uses to be + adjective.

So:

• הוא כועס = he is angry
• literally, something like he angry

Hebrew usually does not use a present-tense word for is in sentences like this. So you do not say a separate word for is here.

Also, כועס changes for gender and number:

• כועס = masculine singular
• כועסת = feminine singular
• כועסים = masculine/mixed plural
• כועסות = feminine plural

Because כועס normally goes with the preposition על when you say who someone is angry at.

So:

• הוא כועס עליי = he is angry at me
• אני כועס עליך = I’m angry at you
• היא כועסת עליו = she is angry at him

This is just the standard pattern in Hebrew. A learner has to memorize כועס על... as a set expression.

עליי means on me / at me / about me, depending on context. In this sentence, it means at me because it follows כועס.

It is made from:

• על = on / about / against / at
• -י = me

So עליי = on me / at me

This kind of attached ending is very common in Hebrew prepositions:

• עליי = on me / at me
• עליך = on you / at you
• עליו = on him / at him
• עליה = on her / at her

In modern unpointed Hebrew, עליי is a very common spelling for עָלַי (alai, on me / at me).

The extra י helps avoid confusion with עלי, which can also be read as עֲלֵי in other contexts, such as in more formal or literary language.

So for learners, the important point is:

• עליי = the normal modern spelling you will often see for on me / at me
• pronunciation: a-LAI

Because it agrees with the speaker, which in this sentence is אני = I.

In Hebrew, even with I, adjectives and participles still show gender:

• אני לא יודע = I don’t know (male speaker)
• אני לא יודעת = I don’t know (female speaker)

So the sentence as written sounds like it is being said by a male speaker.

Only the part referring to I would need to change:

• הוא כועס עליי, אבל אני לא יודעת למה.

Why only that part?

• הוא כועס stays the same because הוא is he
• אני לא יודעת changes because the speaker is female

If the angry person were female, then you would also change the first part:

• היא כועסת עליי, אבל אני לא יודע למה.
• or, if the speaker is also female: היא כועסת עליי, אבל אני לא יודעת למה.

Here למה means why, but the whole sentence is not a direct question. It contains an embedded question or indirect question:

• אני לא יודע למה = I don’t know why

So Hebrew, like English, can place why inside a statement.

Compare:

• Direct question: למה הוא כועס עליי? = Why is he angry at me?
• Indirect question: אני לא יודע למה הוא כועס עליי. = I don’t know why he is angry at me.

In your sentence, the second clause ends with למה by itself, which is also perfectly natural: I don’t know why.

Because the whole sentence is a statement, not a direct question.

The speaker is not asking Why is he angry at me?
The speaker is saying He is angry at me, but I don’t know why.

That is why a period is correct, not a question mark.

Yes, but מדוע is usually more formal.

So:

• אני לא יודע למה = normal, everyday, common
• אני לא יודע מדוע = more formal/literary

Most everyday speech uses למה.

A natural pronunciation is:

Hu ko'es alai, aval ani lo yode'a lama.

A few notes:

• הוא = hu
• כועס = ko'ES
  The little break in the middle comes from the ע, though in fast modern speech it is often weak.
• עליי = a-LAI
• אבל = a-VAL
• אני = a-NI
• לא = lo
• יודע = yo-DE-a
• למה = LA-ma

It could, but that would be a different style and usually less natural here.

The most natural Hebrew is simply:

• אני לא יודע למה = I don’t know why

If you say something like:

• אני לא יודע את הסיבה = I don’t know the reason

that sounds more explicit and a bit less conversational. It is grammatical, but it does not feel as natural in this context as אני לא יודע למה.

Yes, absolutely.

Your sentence is:

• הוא כועס עליי, אבל אני לא יודע למה.

A fuller version could be:

• אני לא יודע למה הוא כועס עליי.

Both are natural. The shorter למה at the end works because the rest is understood from context.

This is very common in Hebrew: if the missing information is obvious, it can be left unstated.