Masayı biraz sola kaydırabilir misin?

Because masa is the direct object of the verb, and here it refers to a specific table: the table.

Turkish often marks a specific direct object with the accusative ending:

• masa = a table / table
• masayı = the table

So:

• Masayı kaydırabilir misin? = Can you move the table?
• Masa kaydırabilir misin? would sound wrong in this context.

Also, the -y- is just a buffer consonant. Since masa ends in a vowel and the accusative ending also begins with a vowel, Turkish inserts y:

• masa + ı → masayı

Biraz means a little / a bit / slightly.

In this sentence, it tells you how much to move the table:

• Masayı biraz sola kaydırabilir misin?
• Can you move the table a little to the left?

It also makes the request sound a bit softer and more natural. Compare:

• Masayı sola kaydırabilir misin? = Can you move the table to the left?
• Masayı biraz sola kaydırabilir misin? = Can you move the table a little to the left?

Because sola means to the left, with a sense of direction.

Here:

• sol = left
• sola = to the left

The ending -a / -e is the dative case, which often shows direction toward something.

So:

• sola = to the left
• solda = on the left / at the left
• soldan = from the left

This sentence needs sola because the table is being moved toward the left.

It can be broken down like this:

• kaydır- = move/slide something
• -abil- = can / be able to
• -ir = a common tense/aspect form used in polite requests and general statements
• mi = question particle
• -sin = you (informal singular)

So:

• kaydırabilir misin? = can you move/slide it?

A rough literal breakdown is:

• kaydırabilir misin?
• move-can-you?

Even if the literal structure looks different from English, this is a very normal Turkish way to say Can you ... ?

Not exactly.

• kaydırmak = to slide, shift, or move something sideways without necessarily lifting it
• taşımak = to carry or transport something

For a table, kaydırmak is very natural if you mean pushing it along the floor a little:

• Masayı biraz sola kaydırabilir misin? = Can you slide the table a little to the left?

If you used taşımak, it would sound more like carry/move the table rather than shift it slightly.

Yes. It is a natural, polite informal request.

...abilir misin? is a common way to say Can you ... ?

Politeness levels:

• Kaydır. = Move it. → direct command
• Kaydırır mısın? = Will you move it? → polite but a bit more direct
• Kaydırabilir misin? = Can you move it? → very common, polite
• Kaydırabilir misiniz? = Can you move it? → polite/formal or plural you

So Masayı biraz sola kaydırabilir misin? is appropriate with a friend, family member, classmate, coworker you are on sen terms with, etc.

Because in Turkish, the question particle mi/mı/mu/mü is usually written separately from the word before it.

Examples:

• geliyor musun? = are you coming?
• biliyor mu? = does he/she know?
• ister misin? = do you want?
• kaydırabilir misin? = can you move it?

But notice that the personal ending attaches to the question particle:

• mi + sin → misin

The form changes by vowel harmony:

• mı, mi, mu, mü

Here it is mi because of the vowels before it.

Because Turkish is usually an SOV language:

• Subject – Object – Verb

In this sentence, the subject you is not stated, because it is already understood from misin.

So the structure is roughly:

• Masayı = the table
• biraz = a little
• sola = to the left
• kaydırabilir misin? = can you move?

That gives:

• [Object] [Adverb] [Direction] [Verb]

This verb-final order is very normal in Turkish.

Yes, Turkish word order is flexible, but not every order sounds equally natural.

The most neutral version here is:

• Masayı biraz sola kaydırabilir misin?

You can move elements around for emphasis, for example:

• Biraz masayı sola kaydırabilir misin?
  This can sound like the emphasis is on a little.
• Masayı sola biraz kaydırabilir misin?
  Possible, but less natural in everyday speech.

For learners, the safest and most natural choice is the original order:

• Masayı biraz sola kaydırabilir misin?

Because Turkish often leaves things understood from context.

In English, you might think of:

• Can you move it a little to the left?

But in Turkish, if the object is named directly, that is enough:

• Masayı biraz sola kaydırabilir misin?

You do not need an extra word for it. Turkish generally does not add a separate object pronoun when the noun is already there.

They both mean you, but they differ in politeness and number.

• misin = you singular, informal
• misiniz = you plural or polite/formal singular

So:

• Masayı biraz sola kaydırabilir misin? = Can you move the table a little to the left?
  → to one person, informal
• Masayı biraz sola kaydırabilir misiniz? = Can you move the table a little to the left?
  → polite/formal, or to more than one person

A few points are useful:

• masayı → ma-sa-yı
  The ı is the Turkish dotless i. It is not exactly like English ee. It is a more central, unrounded vowel.
• biraz → bi-raz
• sola → so-la
• kaydırabilir misin → kay-dı-ra-bi-lir mi-sin

Two especially important sounds:

• ı in kaydırabilir, masayı
• r in Turkish is usually lighter than in most American English accents

You do not need perfect pronunciation immediately, but distinguishing i and ı is very important in Turkish.