Bu ışıltılı takı özel günlerde takılıyor.

Takı is grammatically singular, but in Turkish it can often work like jewelry in English: a general/collective noun.

So Bu ışıltılı takı can mean:

• this sparkling piece of jewelry, if one item is meant
• this sparkling jewelry, if the speaker is talking more generally

Context decides which is intended. If you wanted to clearly emphasize multiple separate items, you could use takılar.

Işıltılı means sparkling, glittering, shiny, radiant.

It is built from:

• ışıltı = sparkle, gleam
• -lı / -li / -lu / -lü = a suffix meaning with, having

So ışıltılı literally means something like with sparkle.

This -lı suffix is very common in Turkish:

• tuzlu = salty, with salt
• sesli = with sound, voiced
• renkli = colorful, with color

In Turkish, adjectives normally come before the noun, just as they do in English.

So:

• ışıltılı takı = sparkling jewelry
• özel günler = special days/occasions

Also, Turkish adjectives do not change form for singular/plural or gender.
So ışıltılı stays the same no matter what noun follows it.

The ending -de is the locative suffix. Here it gives the meaning on / in / during.

So:

• özel günler = special days
• özel günlerde = on special days / on special occasions

In this sentence, özel günlerde tells us when the jewelry is worn.

Turkish often uses gün (day) in expressions where English uses occasion.

So özel günler is a very common phrase meaning:

• special days
• special occasions
• important celebrations

The plural -ler is natural because the sentence is talking about such events in general, not one specific day.

Here takılıyor means is worn or gets worn.

It comes from the verb takmak, which often means:

• to put on an accessory
• to attach
• to fasten

For example:

• kolye takmak = to put on / wear a necklace
• küpe takmak = to put on / wear earrings

In this sentence, the form takılıyor is understood as is being worn / is worn.

This is a very important difference.

• takıyor = he/she/it is putting on or wearing something
• takılıyor = it is being worn / it gets worn

So:

• Kadın takıyı takıyor.
  = The woman is putting on / wearing the jewelry.

• Takı takılıyor.
  = The jewelry is being worn.

In your sentence, the subject is the jewelry, not the person wearing it. So takılıyor is the natural choice.

Yes, in this sentence it functions like a passive form.

Very roughly:

• takmak = to put on / attach
• takılmak = to be put on / to be attached / to be worn

Then:

• takıl-ıyor = is being worn / is worn

Do note that takılmak can also have other meanings in different contexts, such as to get stuck or even to hang out in casual speech. But here, because the subject is takı, the meaning is clearly to be worn.

Turkish -iyor often does mean something happening right now, but it can also be used for:

• repeated behavior
• usual practice
• general present situations

Because of özel günlerde (on special occasions), the sentence is understood as a habitual/general statement:

• Bu ışıltılı takı özel günlerde takılıyor. = This sparkling jewelry is worn on special occasions.

A more timeless or rule-like version could also be takılır, but takılıyor is very natural here.

Bu means this.

So:

• bu takı = this jewelry / this piece of jewelry

Turkish does not have articles exactly like English the and a.
Because of that, words like bu do a lot of work when you want to point something out specifically.

If you remove bu, the sentence becomes more general:

• Işıltılı takı özel günlerde takılıyor. = Sparkling jewelry is worn on special occasions.

With bu, the speaker is pointing to a specific item or type:

• Bu ışıltılı takı... = This sparkling jewelry...

The basic Turkish order here is very normal:

• Bu ışıltılı takı = subject/topic
• özel günlerde = time expression
• takılıyor = verb

So the sentence ends with the verb, which is very typical in Turkish.

You can rearrange parts for emphasis, but the original order is the most neutral:

• Bu ışıltılı takı özel günlerde takılıyor.

For example, you might also hear:

• Özel günlerde bu ışıltılı takı takılıyor.

That puts more focus on özel günlerde.

The letter ı is the dotless i, and it is one of the trickier Turkish sounds for English speakers.

A rough guide:

• i = like the vowel in machine
• ı = a more central, relaxed sound, somewhat like the unstressed vowel in sofa, but not exactly

So ışıltılı is approximately: uh-shuhl-tuh-luh
That is only an approximation, but it helps you avoid pronouncing every ı like English ee.

Also:

• ş = sh
• günlerde has ü, which is like the German ü or French u

A rough pronunciation of the whole sentence is: Boo uh-shuhl-tuh-luh tah-kuh uh-zel goon-lair-deh tah-kuh-luh-yor