Arkadaşım bana yeni projesini anlattı.

Because Turkish usually shows possession with a suffix attached to the noun.

• arkadaş = friend
• arkadaşım = my friend

The ending -ım is the 1st person singular possessive suffix: my.

So:

• arkadaş → friend
• arkadaşım → my friend
• arkadaşın → your friend
• arkadaşı → his/her friend

The exact vowel in the suffix changes because of vowel harmony, which is why it is -ım here.

Because bana is the dative form of ben.

• ben = I
• bana = to me

The verb anlatmak takes the person receiving the information in the dative case.

So:

• Arkadaşım bana anlattı = My friend told me
• literally: My friend told to me

This is very normal in Turkish. Many verbs use case endings where English would just use word order or a preposition.

Also, ben → bana is slightly irregular, just like:

• sen → sana

projesini has several parts:

• proje = project
• projesi = his/her project
• projesini = his/her project (as a definite direct object)

Breakdown:

• proje
• -si = his/her/its possessive suffix
• -ni = accusative ending here, with a buffer n

So projesini means something like his/her project in the role of the thing that was told/explained.

Because Turkish often does not use a separate possessive word like English does. Instead, possession is usually built into the noun with a suffix.

So instead of saying something like his project with a separate word, Turkish says:

• projesi = his/her project

This means the sentence does not need a separate word for his or her.

Also, Turkish o can mean he, she, or it, so the possessive suffix does not tell you gender. projesi can mean:

• his project
• her project
• its project

You understand which one from context.

That n is a buffer consonant.

When Turkish adds certain suffixes after a possessed noun, it often inserts n to help the word flow smoothly.

Here is the sequence:

• proje
• projesi = his/her project
• projesi + i would be awkward
• so Turkish uses n: projesini

This n does not have a meaning by itself. It is there for pronunciation and grammar.

Because Turkish adjectives do not agree with nouns the way they do in many European languages.

• yeni proje = new project
• yeni projeler = new projects
• yeni projesi = his/her new project

The adjective yeni stays the same. It does not change for:

• singular/plural
• gender
• case
• possession

That is much simpler than in many languages.

Because it is the definite direct object of the verb.

In Turkish, a specific or definite direct object usually takes the accusative case.

Here, the project is not just a project in a general sense. It is a specific one: his/her new project. So it appears with the accusative ending.

Compare:

• Bir proje anlattı = He/she explained a project / a project was explained
• Projesini anlattı = He/she explained his/her project

In your sentence, the object is clearly specific, so the accusative is expected.

anlattı is the verb in the simple past tense.

It comes from anlatmak = to tell / to explain.

Breakdown:

• anlat- = tell, explain
• -tı = past tense
• no extra personal ending is needed here for he/she/it

So anlattı means:

• he/she told
• he/she explained

The final form has tt because the verb stem already ends in t, and then the past tense suffix begins with t.

Because Turkish is usually a verb-final language.

A very common Turkish order is:

• Subject + indirect object + direct object + verb

So in this sentence:

• Arkadaşım = subject
• bana = indirect object
• yeni projesini = direct object
• anlattı = verb

That said, Turkish word order is fairly flexible. Speakers can move parts around for emphasis, but the verb often stays at the end.

Yes, in some contexts.

Turkish often leaves out information that is already clear.

For example:

• Bana yeni projesini anlattı. = My friend told me about his/her new project.
  This works if it is already clear who he/she is from context.

Or even:

• Yeni projesini anlattı. = He/she told about his/her new project.
  This works if both the subject and the listener are already understood.

But in your full sentence, Arkadaşım and bana are included to make everything explicit and clear.

It can be either, depending on context.

anlatmak often means:

• to tell
• to explain
• to describe
• to talk about

In this sentence, it can naturally mean something like:

• told me about his/her new project
• explained his/her new project to me

Turkish often uses one verb where English might choose between several different verbs depending on style and context.