Шашлык с чесноком ароматнее, чем без чеснока.

In Russian, the verb to be is normally omitted in the present tense.

So:

• Шашлык с чесноком ароматнее
  literally: Shashlik with garlic more aromatic

means:

• Shashlik with garlic is more aromatic

This is completely normal Russian.
If the sentence were in the past or future, a form of быть would appear.

For example:

• Шашлык был ароматнее = The shashlik was more aromatic
• Шашлык будет ароматнее = The shashlik will be more aromatic

Because с in the meaning with normally requires the instrumental case.

• dictionary form: чеснок
• instrumental singular: чесноком

So:

• с чесноком = with garlic

This is a very common pattern:

• с луком = with onion
• с перцем = with pepper
• с мясом = with meat

So the ending changes not because the word itself changes meaning, but because the preposition с controls the instrumental case.

Because без always takes the genitive case.

• dictionary form: чеснок
• genitive singular: чеснока

So:

• без чеснока = without garlic

This is another very common pattern:

• без соли = without salt
• без масла = without butter/oil
• без сахара = without sugar

So in this sentence you see the same noun in two different cases:

• с чесноком = instrumental
• без чеснока = genitive

Ароматнее is the comparative form of ароматный meaning aromatic, fragrant.

• ароматный = aromatic
• ароматнее = more aromatic

Russian often forms comparatives with endings like:

• -ее
• -ей
• sometimes other changes depending on the adjective

So this sentence means:

• Shashlik with garlic is more aromatic than without garlic

Russian has two common ways to make a comparative:

1. short comparative form

   • ароматнее = more aromatic

2. более + adjective

   • более ароматный = more aromatic

In everyday Russian, the short comparative ароматнее is very natural here.

Чем means than in comparisons.

So:

• ароматнее, чем без чеснока
  = more aromatic than without garlic

A very common Russian pattern is:

• X ...ее, чем Y
  = X is more ... than Y

Examples:

• Это вкуснее, чем то. = This is tastier than that.
• Летом теплее, чем зимой. = It is warmer in summer than in winter.

Here, чем introduces the thing being compared to.

Yes, something is omitted, and that is very natural.

The full idea is:

• Шашлык с чесноком ароматнее, чем шашлык без чеснока.

But Russian often leaves out repeated words when they are obvious from context. So the shorter sentence is preferred:

• Шашлык с чесноком ароматнее, чем без чеснока.

This works just like English:

• Shashlik with garlic is more aromatic than without garlic.

English also omits the repeated noun there.

Шашлык is a very common word for skewered grilled meat, especially in the cuisine of the former Soviet world, the Caucasus, and nearby regions.

In English, depending on context, it may be translated as:

• shashlik
• kebab
• barbecue meat
• grilled skewered meat

Grammatically, it is a masculine singular noun here.

Because it agrees with шашлык, which is a singular masculine noun.

Even if the meaning is somewhat general, grammatically the subject is still one noun:

• шашлык = singular masculine

So the comparative form is built from the adjective describing that noun.
Unlike full adjectives, this comparative form does not change for gender or number:

• ароматнее can be used with masculine, feminine, neuter, or plural nouns

For example:

• суп ароматнее = the soup is more aromatic
• рыба ароматнее = the fish is more aromatic
• блюдо ароматнее = the dish is more aromatic

Yes. A few natural alternatives are possible, for example:

• Шашлык с чесноком более ароматный, чем без чеснока.
• Шашлык с чесноком пахнет лучше, чем без чеснока.
• Шашлык с чесноком ароматнее, чем шашлык без чеснока.

But the original sentence is very natural and concise.

The version with ароматнее is especially idiomatic because Russian often prefers the simple comparative form over более + adjective when possible.

Not completely. Russian word order is flexible, though some orders sound more natural than others.

The original:

• Шашлык с чесноком ароматнее, чем без чеснока.

is the most neutral and natural.

You could also say:

• С чесноком шашлык ароматнее, чем без чеснока.

This puts extra emphasis on with garlic.

So the meaning stays basically the same, but the focus can shift depending on word order.

Because чем introduces the second part of the comparison, and in standard Russian punctuation, it is usually separated by a comma in this structure.

So:

• ароматнее, чем без чеснока

This is the normal written form.

You will often see the same thing in other comparisons:

• Он выше, чем брат. = He is taller than his brother.
• Сегодня холоднее, чем вчера. = Today is colder than yesterday.

So the comma here is expected and standard.