По сравнению со старой квартирой эта кухня шире, но высота шкафов здесь ниже.

По сравнению с/со is a fixed expression meaning compared with, in comparison with.

The word сравнению is the dative singular form of сравнение. That is because the preposition по often takes the dative, and in this expression the whole phrase is frozen as по сравнению с/со....

So structurally you can think of it as:

• по сравнению = in comparison
• с/со ... = with ...

Со is just a pronunciation-friendly variant of с. Russian often uses со before awkward consonant clusters.

Since старой begins with ст-, со старой sounds smoother than с старой.

So:

• с братом
• со старой квартирой

Both с and со mean with here; the difference is mainly phonetic.

Because in the expression по сравнению с/со, the noun after с/со goes in the instrumental.

Base forms:

• старая квартира = old apartment
• со старой квартирой = compared with the old apartment

Both the adjective and the noun change to match the instrumental singular feminine:

• старая → старой
• квартира → квартирой

Because по сравнению с... is usually treated as a normal adverbial phrase, not as something that must be set off by commas.

So По сравнению со старой квартирой эта кухня шире... is perfectly standard without a comma.

A writer could add punctuation in some contexts for emphasis or intonation, but here the version without a comma is the neutral, ordinary one.

Эта кухня is the subject of the clause, so it is in the nominative case.

The structure is basically:

• эта кухня шире = this kitchen is wider

If you used эту кухню, that would be accusative, which would suggest it is a direct object, not the subject.

In the present tense, Russian usually omits the verb быть (to be).

So:

• эта кухня шире literally looks like this kitchen wider
• but it means this kitchen is wider

The same happens in the second clause:

• высота шкафов здесь ниже = the height of the cabinets here is lower

This is one of the most common patterns in Russian.

They are comparative forms:

• широкий → шире = wider
• низкий → ниже = lower

Russian often prefers these short comparative forms instead of более + adjective.

So:

• эта кухня шире = this kitchen is wider
• высота шкафов ниже = the cabinet height is lower

You could sometimes say более широкая or более низкая, but in a sentence like this шире and ниже sound more natural and compact.

Russian does not always need a separate word meaning than.

In the first clause, the comparison is already introduced by по сравнению со старой квартирой. So шире is understood as wider than in the old apartment / than before.

In the second clause, ниже continues the same comparison from context. Russian often leaves the comparison target understood if it is already clear.

If you wanted to state than more directly, Russian could use чем, but it is not necessary here.

Because the sentence focuses specifically on the height as a property.

• высота шкафов здесь ниже = the height of the cabinets here is lower

This is a bit more precise than simply saying:

• шкафы здесь ниже = the cabinets here are lower/shorter

Both can work, but высота шкафов puts the emphasis on the measurement or dimension, not just on the cabinets themselves.

Because высота usually takes a dependent noun in the genitive: the height of something.

So:

• высота чего? = the height of what?
• шкафов = of the cabinets

The dictionary form is шкаф. Its nominative plural is шкафы, but after высота you need the genitive plural шкафов.

Здесь means here, and it anchors the second clause to the current place: here, in this apartment / in this kitchen.

Russian word order is flexible, so you could move it:

• высота шкафов здесь ниже
• здесь высота шкафов ниже

The version in the sentence sounds neutral and natural. Moving здесь changes the emphasis slightly, but not the basic meaning.

Yes, it can sound slightly compressed in English too. The idea is understood as something like:

• Compared with the old apartment, this kitchen is wider
• meaning compared with the kitchen in the old apartment

Russian often allows this kind of shorthand when the context makes the intended comparison obvious.

If you wanted to make it fully explicit, you could say something like:

По сравнению с кухней в старой квартире эта кухня шире...

That is more precise, but the original sentence is still perfectly understandable.