Мне нравится та аллея, где весной цветёт роза и где почти не видно паутины.

Мне нравится... literally works like To me, ... is pleasing.

So in this pattern:

• мне = to me (dative case)
• нравится = is pleasing / appeals
• the thing being liked is the grammatical subject

That is why аллея is the thing that нравится.

You could say Я люблю ту аллею, but that is a bit stronger and more direct: I love that alley.
Мне нравится та аллея is often a more neutral I like that alley.

Because нравится agrees with the thing being liked, and here that thing is та аллея, which is:

• singular
• feminine

The verb form is singular because аллея is singular.

Compare:

• Мне нравится эта аллея. = I like this alley.
• Мне нравятся эти аллеи. = I like these alleys.

So the verb does not agree with мне. It agrees with аллея.

Мне is the dative case of я.

In the structure кому-то нравится что-то, the person who experiences the feeling goes into the dative:

• мне нравится = I like
• тебе нравится = you like
• ему нравится = he likes
• нам нравится = we like

So мне here means to me.

Because та аллея is the subject of нравится, so it stays in the nominative case.

• та = nominative feminine singular of тот
• аллея = nominative singular

Compare:

• Мне нравится та аллея.
  Here та аллея is the thing that is pleasing, so nominative.

• Я люблю ту аллею.
  Here ту аллею is the direct object of люблю, so accusative.

This is a very common point for English speakers, because in English both sentences would simply use that alley.

Та is the feminine singular form of тот, meaning that.

Because аллея is feminine, the demonstrative also has to be feminine:

• тот сад = that garden
• та аллея = that alley
• то дерево = that tree
• те улицы = those streets

So та аллея means that alley.

Here где means where and introduces a clause describing аллея.

So:

• та аллея, где весной цветёт роза...
  = that alley where a rose blooms in spring...

Russian often uses где to connect a place noun with a relative clause.

Other examples:

• дом, где я живу = the house where I live
• парк, где гуляют дети = the park where children walk/play
• улица, где много магазинов = the street where there are many shops

Because there are really two parallel descriptive clauses attached to аллея:

• где весной цветёт роза
• и где почти не видно паутины

Repeating где makes the structure very clear and balanced.

Russian can sometimes omit the second где if the meaning is obvious, but repeating it is natural and often stylistically smoother, especially in written language.

So this sounds like:

• that alley where roses bloom in spring and where you can hardly see any cobwebs

Весной is the instrumental case of весна, and Russian often uses the instrumental to express time in the sense of in spring, in summer, etc.

So:

• весной = in spring
• летом = in summer
• осенью = in autumn
• зимой = in winter

This is a set pattern that learners usually just memorize.

Compare:

• Весна красивая. = Spring is beautiful.
  Here весна is nominative.

• Весной здесь тихо. = It is quiet here in spring.
  Here весной is the adverbial time expression.

Цветёт is the 3rd person singular present-tense form of цвести, meaning to bloom / to flower.

It is singular because the subject is роза:

• роза цветёт = the rose blooms

Some related forms:

• я цвету
• ты цветёшь
• он / она цветёт
• они цветут

So in this sentence, цветёт simply agrees with роза.

Because the sentence presents a single rose as blooming there: цветёт роза = a rose blooms / the rose blooms.

Russian has no articles, so роза can mean:

• a rose
• the rose

depending on context.

If it were plural, you would expect:

• цветут розы = roses bloom / the roses bloom

So the singular verb цветёт and singular noun роза go together.

This is an impersonal construction.

• видно means visible / can be seen
• не видно means not visible / cannot really be seen
• почти не видно means hardly visible / you can hardly see

So:

• почти не видно паутины
  = there is hardly any spiderweb visible
  = you can hardly see any cobweb

There is no normal subject like in English. Russian often uses these impersonal words:

• видно = visible / can be seen
• слышно = audible / can be heard
• жарко = hot
• холодно = cold

Because after не видно, Russian very often uses the genitive case for the thing that is absent or not perceptible.

So:

• паутина = nominative
• паутины = genitive singular

In не видно паутины, the idea is roughly no spiderweb is visible or there is almost no spiderweb to be seen.

This use of the genitive is very common after negation and in expressions of absence:

• нет времени = there is no time
• не видно дома = the house cannot be seen
• не слышно музыки = no music can be heard

It can do both, depending on context.

In this sentence, паутины is best understood as spiderweb / cobweb in a more general or mass-like sense: there is hardly any spiderweb visible there.

Russian often uses singular nouns this way when talking about a substance or a general presence of something.

So a natural English rendering would usually be:

• where there is hardly any cobweb
• where you can hardly see any spiderwebs

Both are reasonable depending on style.

The word order is natural, but Russian word order is more flexible than English.

This sentence is arranged in a fairly neutral way:

• Мне нравится = sets up the personal reaction
• та аллея = names the thing liked
• then the two где clauses describe it

Inside the clauses:

• весной цветёт роза = in spring a rose blooms
• почти не видно паутины = hardly any spiderweb is visible

You could shift words around for emphasis, but the given order sounds smooth and standard.

A very literal version would be something like:

To me pleases that alley where in spring a rose blooms and where almost no spiderweb is visible.

That sounds unnatural in English, but it helps show the Russian structure:

• Мне = to me
• нравится = pleases / is pleasing
• та аллея = that alley
• где... = where...
• почти не видно паутины = almost no spiderweb is visible

A natural English translation would be something more like:

I like that alley where a rose blooms in spring and where you can hardly see any cobwebs.

It is grammatically normal Russian, but it has a slightly descriptive, written feel because of the parallel structure:

• где весной цветёт роза
• и где почти не видно паутины

The repetition of где and the use of не видно паутины give it a polished, somewhat literary tone.
It is not unnatural in everyday Russian, but it sounds more carefully phrased than casual speech.