Domaćin objašnjava pravilo saigraču.

• Domaćin = the subject, the person doing the action
• objašnjava = the verb, explains / is explaining
• pravilo = the direct object, the thing being explained
• saigraču = the indirect object, the person receiving the explanation, so to the teammate

So the structure is basically:

[subject] + [verb] + [thing explained] + [person it is explained to]

Because it is in the dative case.

In Serbian, the person something is given, shown, explained, said, etc. to is often put in the dative.
Here, saigrač means teammate, and saigraču means to the teammate.

Compare:

• saigrač = teammate
• saigraču = to the teammate

The verb objašnjavati / objasniti commonly takes:

• a direct object in the accusative: pravilo
• an indirect object in the dative: saigraču

So:

• objašnjava pravilo = explains the rule
• objašnjava saigraču = explains to the teammate

Together:

• objašnjava pravilo saigraču = explains the rule to the teammate

It actually is the direct object, but with this noun the nominative and accusative forms are the same.

Pravilo is a neuter singular noun. For many neuter singular nouns in Serbian, the nominative and accusative look identical.

So:

• pravilo = rule (subject form)
• pravilo = rule (direct object form)

Even though the form does not change, its function does:

• in this sentence, pravilo is in the accusative because it is the thing being explained

Objašnjava is present tense, 3rd person singular.

It means:

• he explains
• she explains
• it explains
• depending on context, also is explaining

Serbian present tense often covers both simple present and present continuous meanings that English separates.

So Domaćin objašnjava pravilo saigraču can mean:

• The host explains the rule to the teammate
• The host is explaining the rule to the teammate

Objašnjava is from the imperfective verb objašnjavati / objašnjavati?
The usual pair is:

• objašnjavati = imperfective, ongoing/repeated explaining
• objasniti = perfective, explain fully / complete the explanation

In practice, objašnjava comes from the imperfective verb and suggests:

• an ongoing action
• a repeated or habitual action
• focus on the process, not the completed result

So this sentence sounds like:

• The host is explaining the rule to the teammate or
• The host explains the rule to the teammate

If you used a perfective form, the meaning would shift toward completing the explanation.

Because Serbian does not have articles like English a/an and the.

So:

• domaćin can mean a host or the host
• pravilo can mean a rule or the rule
• saigraču can mean to a teammate or to the teammate

The exact meaning depends on context.

That is why one Serbian sentence can correspond to several natural English translations.

Yes. Serbian word order is more flexible than English because case endings show grammatical roles.

The neutral order here is:

Domaćin objašnjava pravilo saigraču.

But other orders are possible, for example:

• Domaćin saigraču objašnjava pravilo.
• Pravilo domaćin objašnjava saigraču.
• Saigraču domaćin objašnjava pravilo.

These all keep roughly the same core meaning, but they can change emphasis or information focus.

English usually has to keep a stricter order, while Serbian can move words around more freely.

Yes. Domaćin has a wider range of meanings than English host.

Depending on context, it can mean:

• host
• master of the house / head of household
• home side / home team in sports
• sometimes more generally a person acting as a host or organizer

So in some contexts, especially sports, domaćin might not mean a person hosting guests literally.
It could refer to the home player, home team, or someone on the hosting side.

A rough English-friendly guide:

• Domaćin ≈ doh-MAH-chin
  
  • ć is a soft sound, somewhat like a very soft ch/t-y
• objašnjava ≈ ob-YASH-nya-va
  • š = sh
  • nj = ny as in canyon
• pravilo ≈ PRAH-vee-lo
• saigraču ≈ sigh-GRAH-choo
  • č is a harder ch sound

A more important pronunciation note:

• Serbian spelling is highly phonetic
• j is pronounced like English y
• š = sh
• č = ch
• ć = a softer version of č

You would normally use a past tense form, for example:

Domaćin je objašnjavao pravilo saigraču.
or
Domaćin je objasnio pravilo saigraču.

The difference is aspect:

• je objašnjavao = was explaining / explained over some period of time
• je objasnio = explained completely / finished explaining

So if you want a simple completed action, objasnio is often the better choice.

It is the normal singular dative form of saigrač, yes.

Useful related forms are:

• saigrač = teammate
• saigrača = teammate (some other cases, depending on function)
• saigraču = to the teammate
• saigračem = with the teammate

So in this sentence, saigraču is exactly the form needed because the verb is directed to that person.

If the sentence were plural, you would need a different form:

• saigračima = to the teammates

So:

• Domaćin objašnjava pravilo saigračima. = The host is explaining the rule to the teammates.