L'indovinello dà un indizio.

Because indovinello starts with a vowel.

In Italian, the singular definite article becomes l' before a vowel:

• l'indovinello = the riddle
• l'amico = the friend
• l'idea = the idea

The apostrophe shows that a vowel was dropped from the article.

Indovinello is a masculine singular noun.

You can tell it is masculine here because it takes the masculine article l' and would normally pattern like other masculine nouns:

• singular: l'indovinello
• plural: gli indovinelli

The ending -o is also a very common masculine ending in Italian, though not every noun follows that pattern.

Yes. Dà is the third-person singular present of dare, meaning to give.

So here:

• L'indovinello = the riddle
• dà = gives
• un indizio = a clue

So the structure is literally The riddle gives a clue.

Other present-tense forms of dare are:

• do = I give
• dai = you give
• dà = he/she/it gives
• diamo = we give
• date = you all give
• danno = they give

The accent is used to distinguish the verb dà from the very common preposition da, which can mean from, by, or at someone's place, depending on context.

So:

• dà = gives
• da = from / by / since / at

For a learner, it is very useful to keep writing dà with the accent so the meaning stays clear.

Because indizio starts with a vowel.

For masculine singular nouns, the indefinite article is:

• un before most words, including words beginning with a vowel
• uno before certain consonant sounds such as z, s + consonant, ps, gn, and a few others

So:

• un indizio
• un amico
• uno studente
• uno zaino

That is why un indizio is correct.

Because this is a masculine noun.

With masculine nouns, un is written without an apostrophe, even before a vowel:

• un indizio
• un amico

The apostrophe appears with the feminine article una, which becomes un' before a vowel:

• un'idea
• un'amica

So:

• un indizio = correct
• un'indizio = incorrect

It is a normal subject + verb + object sentence:

• L'indovinello = subject
• dà = verb
• un indizio = direct object

This is the most neutral and straightforward word order in Italian, just like in English:

• The riddle gives a clue.

Italian can change word order for emphasis, but this sentence is in the standard, unmarked order.

Because un indizio is the direct object of dare here.

In this sentence, the riddle is giving something: a clue. That something is the direct object, so no preposition is needed.

Compare:

• L'indovinello dà un indizio. = The riddle gives a clue.
• L'indovinello dà un indizio al lettore. = The riddle gives a clue to the reader.

Here:

• un indizio = what is given
• al lettore = to whom it is given

So Italian uses a only for the recipient, not for the thing being given.

Normally, no.

In standard Italian, common nouns usually need an article or another determiner when they are used as the subject of a sentence. So L'indovinello dà un indizio sounds normal, but Indovinello dà un indizio sounds incomplete or unnatural in ordinary prose.

Without the article, it might sound like:

• a title
• a note
• a dictionary-style label
• very telegraphic language

So for normal sentence use, keep the article.

Not in the way a learner might expect.

They may look slightly similar at first glance, but they mean different things:

• indovinello = riddle
• indizio = clue, hint, sign

So in this sentence, the idea is that the riddle provides a clue.

It is worth learning them as two separate vocabulary items.

A simple English-friendly approximation is:

leen-doh-vee-NEL-loh dah oon een-DEE-tsyoh

A few useful pronunciation notes:

• gli is not in this sentence, so no special issue there.
• ll in indovinello is clearly pronounced.
• z in indizio sounds like ts here.
• The stress falls on:
  • indovinELlo
  • indIzio

A more IPA-style version is approximately:

lin-dovi-NEL-lo da un in-DI-tsyo