Geç kaldığım için biraz mahcup oldum, ama ablam bana gülümseyince rahatladım.

Geç kalmak is a fixed expression meaning to be late.

• geç = late
• kalmak = to remain / stay

But together, geç kalmak does not literally mean to stay late here. It means to arrive late or to be late.

So:

• Geç kaldım = I was late / I arrived late

Because için usually needs to follow a noun or a noun-like clause, not a normal finite verb like kaldım.

So Turkish turns I was late into a clause meaning something like:

• kaldığım = the fact that I was late / my being late

Then:

• Geç kaldığım için = because I was late

This is a very common Turkish pattern:

• geldiğim için = because I came
• yorgun olduğum için = because I am/was tired
• seni sevdiğim için = because I love you

Kaldığım can be broken down like this:

• kal- = verb stem
• -dık / -dik / -duk / -dük = a clause-forming suffix
• -ım = my / I marker

In this sentence:

• kaldığım = that I was late / my being late

The form changes because of vowel harmony and sound changes, so you see -dığım instead of a simple dictionary-style ending.

A useful way to think of it is:

• kaldım = I was late
• kaldığım = that I was late

Literally, it is close to:

• biraz = a little / somewhat
• mahcup = embarrassed, ashamed, sheepish
• oldum = I became

So biraz mahcup oldum is literally I became a little embarrassed.

In natural English, that can be:

• I felt a little embarrassed
• I was a bit embarrassed

Turkish often uses adjective + olmak to mean to become / to feel a certain way.

For example:

• mutlu oldum = I became happy / I was happy
• üzgün oldum = I became sad / I felt sad
• mahcup oldum = I felt embarrassed

Mahcup means something like:

• embarrassed
• ashamed
• sheepish
• sometimes self-conscious

In this sentence, it probably means embarrassed because the speaker was late.

It is a bit softer and more personal than a purely formal word for shame. It often suggests a feeling like I felt bad and awkward.

Because ablam means my older sister.

• abla = older sister
• ablam = my older sister

The -m is the 1st person possessive suffix: my.

So:

• abla = older sister
• ablan = your older sister
• ablası = his/her older sister
• ablam = my older sister

In this sentence, the subject of the second part is explicitly my older sister, so Turkish uses ablam.

Because gülümsemek takes the dative case for the person you smile at.

So:

• ben = I
• bana = to me
• bana gülümsemek = to smile at me

That is why the sentence says:

• ablam bana gülümseyince = when my older sister smiled at me

It is similar to other Turkish verbs that use the dative:

• bana baktı = she looked at me
• bana yardım etti = she helped me
• bana güldü = she laughed at me

So with gülümsemek, bana is the normal choice.

Gülümseyince means:

• when (she) smiled
• once (she) smiled
• sometimes as soon as (she) smiled

It comes from:

• gülümsemek = to smile
• -ince / -ınca = when / once

Because the verb stem ends in a vowel, a buffer y appears:

• gülümse + yince → gülümseyince

So:

• ablam bana gülümseyince = when my older sister smiled at me

This ending often gives a sense of one thing happening, and then another thing happens as a result or immediately after.

Because the subject is already given as ablam right before it.

In Turkish, a form like gülümseyince does not have to carry a separate person ending in this kind of structure if the subject is clear from context or explicitly stated.

So:

• ablam bana gülümseyince = when my older sister smiled at me

The noun ablam already tells you who did the smiling.

Rahatladım comes from rahatlamak, which means:

• to relax
• to feel relieved
• to become at ease

So rahatladım means:

• I relaxed
• I felt relieved
• I became more comfortable / at ease

In this sentence, I felt relieved is probably the best natural English meaning, because the speaker was embarrassed at first, but then felt better when the older sister smiled.

The sentence is:

Geç kaldığım için biraz mahcup oldum, ama ablam bana gülümseyince rahatladım.

A literal breakdown is:

• Geç kaldığım için = because I was late
• biraz mahcup oldum = I felt a little embarrassed
• ama = but
• ablam bana gülümseyince = when my older sister smiled at me
• rahatladım = I felt relieved

So the whole flow is:

1. reason: because I was late
2. result: I felt a little embarrassed
3. contrast/change: but
4. new event: when my older sister smiled at me
5. final result: I felt relieved

This is very natural Turkish word order: the background information and subordinate clauses often come before the main verb.