הגעתי כמעט בזמן, אבל הפגישה כבר התחילה.

In Hebrew, the verb itself usually tells you who the subject is.
הגעתי means I arrived all by itself.

The ending -תי is the past-tense ending for I in many verbs.

So:

• הגעתי = I arrived
• אני הגעתי = I arrived / I did arrive

Adding אני is possible, but it usually adds emphasis.

הגעתי is the past tense, first person singular form of the verb להגיע (to arrive, to reach).

Breakdown:

• להגיע = to arrive
• הגעתי = I arrived

This verb is very common in everyday Hebrew:

• הגעתי הביתה = I arrived home
• הגעתי מוקדם = I arrived early
• הגעתי כמעט בזמן = I arrived almost on time

Literally, כמעט means almost, and בזמן means on time or in time.

So:

• כמעט = almost
• בזמן = on time / in time

Together:

• כמעט בזמן = almost on time

In natural English, that often implies a little late.

This is just the normal Hebrew expression.
Hebrew says בזמן, literally something like in time, where English often says on time.

Examples:

• אני בזמן = I am on time
• הוא לא הגיע בזמן = He didn’t arrive on time
• הגענו בדיוק בזמן = We arrived exactly on time

So even though English uses on, Hebrew uses the preposition ב־ here.

אבל means but.

It connects two ideas that contrast with each other:

• הגעתי כמעט בזמן = I arrived almost on time
• אבל הפגישה כבר התחילה = but the meeting had already started

So the speaker is saying: I was close to being on time, but it was still too late.

The ה־ at the beginning is the Hebrew definite article, meaning the.

So:

• פגישה = a meeting
• הפגישה = the meeting

In this sentence, we are talking about a specific meeting already known from context, so Hebrew uses הפגישה.

Because הפגישה (the meeting) is a feminine singular noun, and in the past tense Hebrew verbs agree with the subject in gender and number.

So:

• הפגישה התחילה = the meeting started
  feminine singular verb
• הסרט התחיל = the movie started
  masculine singular verb

That is why you get:

• התחילה with פגישה
• not התחיל

No. In התחילה, the first ה is part of the verb form itself, not the definite article.

Compare:

• הפגישה = the meeting
  here ה־ really means the
• התחילה = started
  here the ה is just part of the past-tense verb form

So even though both words begin with ה, they are doing different jobs.

כבר means already.

In this sentence:

• הפגישה כבר התחילה = the meeting already started / had already started

Its position is very natural in Hebrew. It often comes before the verb it modifies:

• כבר התחילה = already started
• כבר אכלתי = I already ate
• כבר ראינו את זה = we already saw that

Hebrew word order is somewhat flexible, but כבר before the verb is very common.

Hebrew often uses the simple past where English may prefer the past perfect.

So:

• הפגישה כבר התחילה literally looks like the meeting already started
• but in natural English, because it happened before the speaker arrived, we often say
  the meeting had already started

The word כבר helps create that sense of it was already true by that time.

Yes, Hebrew has some flexibility, but the given version is very natural:

• הגעתי כמעט בזמן, אבל הפגישה כבר התחילה.

You may also hear slightly different emphasis patterns, such as:

• הגעתי כמעט בזמן, אבל כבר הפגישה התחילה.

But that sounds less neutral and less standard for everyday learning.
The original sentence is the best model to copy.

A common pronunciation would be:

higáti kim'át bazmán, avál hapgishá kvar hitchíla

Rough guide:

• הגעתי = hi-GA-ti
• כמעט = kim-AT
• בזמן = baz-MAN
• אבל = a-VAL
• הפגישה = hap-gi-SHA
• כבר = kvar
• התחילה = hit-CHI-la

The stress usually falls near the end in these words, especially:

• הגעתי
• כמעט
• בזמן
• הפגישה
• התחילה