Meine Nachhilfelehrerin meint, man solle lieber langsam sprechen, damit die Aussprache natürlicher klinge.

Nachhilfelehrerin means female tutor or female private teacher.

It is a compound word:

• Nachhilfe = tutoring, extra help with schoolwork
• Lehrerin = female teacher

So meine Nachhilfelehrerin is my female tutor. If it were a man, it would be mein Nachhilfelehrer.

Meinen often means to think, to believe, or to be of the opinion.

So Meine Nachhilfelehrerin meint ... is something like:

• My tutor thinks ...
• My tutor says/maintains that ...
• My tutor is of the opinion that ...

It is a little more about her opinion or judgment than plain sagt would be.

Here, man means one, people, or informal English you in a general sense.

It does not mean man as in adult male.

So:

• man solle lieber langsam sprechen
  means
• one should rather speak slowly
• or more naturally in English, you should speak slowly

A useful point: man always takes a singular verb.

Solle is Konjunktiv I, which is very commonly used for reported speech or indirect speech.

So the sentence is not giving the tutor’s words directly. It is reporting her view:

• direct idea: Sprich lieber langsam.
• reported version: Meine Nachhilfelehrerin meint, man solle lieber langsam sprechen.

Comparison:

• soll = normal indicative, is supposed to / should
• sollte = usually Konjunktiv II, often used for a softer should
• solle = Konjunktiv I, often used when reporting what someone says or thinks

So solle helps create the meaning my tutor says/thinks that one should ...

German can report speech in two common ways here:

1. without dass

   • Meine Nachhilfelehrerin meint, man solle lieber langsam sprechen.

2. with dass

   • Meine Nachhilfelehrerin meint, dass man lieber langsam sprechen solle.

Both are possible.

Without dass, the reported clause has a more main-clause-like word order, so the finite verb solle comes early.

With dass, the finite verb goes to the end of the clause, as usual in a dass clause.

Also notice:

• solle is the finite modal verb
• sprechen stays in the infinitive at the end

That is normal with modal verbs.

Here lieber means rather or preferably.

So:

• lieber langsam sprechen = to speak slowly rather / preferably speak slowly

It comes from the comparative form used with gern:

• gern = gladly / willingly
• lieber = rather / preferably
• am liebsten = most preferably

So lieber does not mean dearer here. It expresses preference.

Damit introduces a purpose clause, meaning so that or in order that.

So:

• damit die Aussprache natürlicher klinge
  means
• so that the pronunciation sounds more natural

This tells you the purpose of speaking slowly.

A useful comparison:

• weil = because
• damit = so that

So this sentence is not explaining a cause; it is explaining a goal or intended result.

Not naturally here, because um ... zu is mainly used when the subject is the same in both parts.

Here the subjects are different:

• man = the person speaking slowly
• die Aussprache = the pronunciation

So damit is the right choice.

Compare:

• Ich lerne viel, um die Prüfung zu bestehen.
  Same subject: ich learn / ich pass.
• Ich spreche langsam, damit die Aussprache natürlicher klingt.
  Different focus in the second clause, so damit works better.

Klinge is again Konjunktiv I.

It continues the reported-speech style started by solle. In other words, the whole idea is still being presented as the tutor’s view.

So:

• klinge = reported / indirect style
• klingt = plain indicative, more direct statement of fact

In practice, many modern speakers might also say klingt, especially in less formal language. But klinge fits well in careful written German because it keeps the sentence consistently in indirect speech.

Natürlicher is the comparative of natürlich:

• natürlich = natural / naturally
• natürlicher = more natural / more naturally

German adjectives are often used as adverbs without adding a special ending like English -ly.

So:

• Die Aussprache klingt natürlich. = The pronunciation sounds natural.
• Die Aussprache klingt natürlicher. = The pronunciation sounds more natural.

Natürlicherweise is a different word and usually means something more like naturally, of course in the sense of as one would expect, not in a more natural-sounding way. So it would not fit well here.