Właścicielka powiedziała, że kawalerka jest umeblowana i bardzo jasna.

Because właścicielka means female owner / landlady. In Polish, nouns often show gender clearly.

• właściciel = male owner
• właścicielka = female owner

So the sentence tells us that the person speaking was a woman.

In the past tense, Polish verbs agree with the gender of the subject.

• powiedział = he said
• powiedziała = she said

Since właścicielka is feminine, the verb must also be feminine: powiedziała.

Polish often leaves out subject pronouns when they are clear from context or from the verb form.

So instead of saying Ona powiedziała..., Polish very naturally says just Powiedziała...

In this sentence, the subject is already given as właścicielka, so adding ona would be unnecessary.

Że means that and introduces a subordinate clause.

So:

• Właścicielka powiedziała = The landlady said
• że kawalerka jest umeblowana i bardzo jasna = that the studio apartment is furnished and very bright

It works much like English that in She said that...

Kawalerka is a studio apartment or bedsit.

It is a feminine noun, which is why the describing words in the sentence are also feminine:

• umeblowana
• jasna

Those forms match kawalerka.

Because umeblowana has to agree with kawalerka, which is feminine singular.

Compare:

• pokój jest umeblowany = the room is furnished
• kawalerka jest umeblowana = the studio apartment is furnished

The ending changes to match grammatical gender.

For the same reason as umeblowana: adjectives in Polish must agree with the noun they describe.

Since kawalerka is feminine singular, we get:

• jasny = masculine form
• jasna = feminine form
• jasne = neuter form

So bardzo jasna means very bright, with the adjective matching kawalerka.

In this sentence, it functions like an adjective: furnished.

Structurally, it comes from a participle, but for a learner it is easiest to understand jest umeblowana as is furnished.

So:

• jest = is
• umeblowana = furnished

Together: is furnished.

It is in the nominative case because it is the subject of the clause:

kawalerka jest umeblowana i bardzo jasna

Here, kawalerka is the thing that is furnished and bright, so nominative is used.

In the present tense, Polish normally uses jest for is.

So:

• kawalerka jest umeblowana = the studio is furnished

Unlike Russian, for example, standard Polish does not usually drop the present-tense verb to be in sentences like this.

Yes, Polish word order is fairly flexible, though the original version is the most neutral and natural.

Standard order: Właścicielka powiedziała, że kawalerka jest umeblowana i bardzo jasna.

You could rearrange parts for emphasis, but the meaning stays similar because endings carry a lot of grammatical information.

For example: Właścicielka powiedziała, że kawalerka jest bardzo jasna i umeblowana.

That is also possible, but it shifts the emphasis slightly.

A rough guide:

• że sounds like zheh, with ż like the s in measure
• właścicielka is roughly vwash-chee-CHYEL-ka, though English spelling can only approximate it

A few useful pronunciation notes:

• ł sounds like English w
• ś is a soft sh-like sound
• cie in -ciciel- is pronounced softly, roughly che
• ż is like the sound in measure

Not with exactly the same meaning.

• mieszkanie = apartment / flat in general
• kawalerka = studio apartment specifically

So if you want to keep the idea of a one-room flat, kawalerka is the right word. Also note that mieszkanie is neuter, so the adjective forms would change:

• mieszkanie jest umeblowane i bardzo jasne

This is a useful example of adjective agreement changing with gender.