Vy říkáte, že se vám líbí naše vesnice, i když je malá.

In Czech the personal pronoun (já, ty, on, vy etc.) is usually dropped because the verb ending already shows the person.
So Říkáte, že… is completely correct and normal.
Here Vy říkáte adds emphasis on you (as opposed to someone else), or can sound slightly more formal/insistent, depending on context.

Že is a conjunction meaning that in reported speech.
So říkáte, že… = you say that….
It introduces the content of what is being said.

Czech uses the verb líbit se in an impersonal, reverse way compared to English.
Literally, (something) líbí se (to someone) = (something) is pleasing to (someone).
So naše vesnice se vám líbí literally means our village is pleasing to you, which in natural English is you like our village.

With líbit se, the person who experiences liking something is in the dative case.
So mně / tobě / vám / mu / jí… are used, not the nominative já / ty / vy / on / ona.
In this sentence, vám = to you (either plural “you all” or polite singular).

Se is part of the verb líbit se; you normally can’t just drop it.
You can think of líbit se as a single verb meaning to be pleasing or to appeal to someone.
The reflexive se also appears when we reverse the roles: Vy se mi líbíte = I like you (literally: you are pleasing to me).

In Czech, short unstressed words like se, mi, ti, vám (called clitics) have preferred positions in the sentence.
In this pattern, se usually comes very early in the clause, and other clitics like mi / ti / nám / vám tend to follow it: že se vám líbí.
Other orders like že vám se líbí sound unnatural or wrong to native speakers in standard Czech.

Yes, mít rád / mít ráda / mít rádi is another common way to say to like.
Vy říkáte, že máte rádi naši vesnici is correct and means essentially the same thing.
However, líbit se is often a bit more neutral or objective (“it appeals to you / you find it nice”), while mít rád can feel slightly more personal or emotional.

Vesnice is a feminine noun in Czech.
The possessive pronoun náš (“our”) has to agree in gender and number with the noun, so the correct feminine singular form is naše.
So: náš dům (masculine), naše vesnice (feminine), naše město (neuter).

Když by itself usually means when.
I když is a fixed phrase meaning even though / although, introducing a contrast or concession.
So i když je malá = even though it is small; it does not mean “when it is small” here.

In Czech, subordinate clauses are usually separated by a comma from the main clause.
I když je malá is a subordinate concessive clause (“even though it is small”), so it’s preceded by a comma.
Writing …naše vesnice i když je malá without the comma would be considered incorrect punctuation in standard writing.

The adjective must agree with the noun it refers to in gender, number, and case.
Vesnice is feminine singular, so the predicate adjective also takes the feminine singular form: malá.
Malý is masculine, so that would agree with a masculine noun like dům (dům je malý = “the house is small”).

Czech almost always drops subject pronouns when they are clear from context or from the verb ending.
So i když je malá is the normal way to say even though it is small, with it understood as vesnice.
I když ona je malá is possible for strong emphasis (something like “even though it IS small”), but in ordinary speech it sounds a bit heavy or contrastive.

Yes. Vy and vám can be either plural “you (all)” or polite singular “you” when addressing one person formally.
So this sentence can be used politely to one person, or neutrally to a group.
Informally to one friend, you would more likely say: Ty říkáš, že se ti líbí naše vesnice, i když je malá.