Bilmece gizemin anahtarını veriyor.

Because the neutral Turkish word order is usually Subject–Object–Verb.

So here you have:

• Bilmece = subject
• gizemin anahtarını = object
• veriyor = verb

English usually prefers Subject–Verb–Object, so Turkish can feel reversed at first.

A few clues show that Bilmece is the subject:

• it is unmarked, which is normal for Turkish subjects
• it comes before the object and verb in the neutral order
• anahtarını is marked as a definite direct object

So Bilmece is the thing doing the action, and gizemin anahtarını is the thing being given.

The ending -in is the genitive ending, which often corresponds to English of or 's.

• gizem = mystery
• gizemin = of the mystery / the mystery's

Turkish commonly uses this in possessive-style noun phrases:

• gizemin anahtarı = literally the mystery's key
• natural English = the key to the mystery

It contains several layers:

• anahtar = key
• anahtar-ı = its key / the key of it
• anahtar-ı-n-ı = its key, as a definite direct object

In this sentence:

• gizem-in marks the possessor
• anahtar-ı marks the possessed noun
• the final -ı marks the whole thing as the definite object of the verb
• the n is a buffer consonant between suffixes

So gizemin anahtarını is structurally something like the mystery's key + object marking.

Because anahtarı is the basic possessed form: its key / the key of it.

But in this sentence, that noun phrase is also the direct object of veriyor, so it needs an extra accusative ending. That produces:

• anahtar-ı = its key
• anahtar-ı-n-ı = its key, as the object

So:

• gizemin anahtarı = the key to the mystery
• gizemin anahtarını = the key to the mystery, as the thing being given

Because Turkish often expresses this kind of relationship with a genitive + possessed noun structure rather than an English-style to phrase.

So English says:

• the key to the mystery

But Turkish naturally says:

• gizemin anahtarı
• literally the mystery's key

This is very common with nouns like key, answer, door, end, and many others.

veriyor comes from:

• ver- = give
• -iyor = present continuous / progressive

So literally it means is giving.

In English, though, this form can sometimes be translated as simple gives, depending on context. Turkish -iyor is often used for actions happening now, but it can also sound vivid and descriptive.

Both are possible Turkish forms, but they have different feels:

• veriyor = is giving / gives in a more immediate, ongoing, or vivid way
• verir = gives in a more general, habitual, or timeless way

So veriyor makes the sentence sound more like an active description, while verir would sound more like a general statement or rule.

Turkish does not have articles like English the and a/an.

Instead, definiteness is usually shown by:

• context
• word order
• case marking

In this sentence, the object is definite, and that is shown by the accusative marking in anahtarını.
Bilmece itself can mean a riddle or the riddle, depending on context, so the English translation has to choose what sounds natural.

Yes. Turkish word order is flexible, but Bilmece gizemin anahtarını veriyor is the neutral version.

If you change the order, you usually change the emphasis:

• Gizemin anahtarını bilmece veriyor = emphasizes bilmece
• Bilmece veriyor gizemin anahtarını = more marked, less neutral

The case endings help keep the sentence understandable even when the order changes.