Hún reið heim á hestinum.

Reið is the past tense, 3rd person singular form of the verb ríða, which means to ride.

So:

• ríða = to ride
• hún ríður = she rides / she is riding
• hún reið = she rode

This is a strong verb, so the past tense is formed by a vowel change rather than by adding a regular ending.

Because heim already means homeward / to home. It is an adverb of direction.

This works a lot like English go home, where English also does not need to:

• fara heim = go home
• koma heim = come home
• ríða heim = ride home

So in Hún reið heim, the word heim already includes the idea of movement toward home.

This is a very common and important distinction:

• heim = home, toward home (direction)
• heima = at home (location)

So:

• Hún reið heim. = She rode home.
• Hún var heima. = She was at home.

In your sentence, heim is used because she is moving toward home.

Here á means on. The phrase á hestinum literally means on the horse.

In English, we might say:

• on the horse
• or more naturally, depending on context, on horseback

So Hún reið heim á hestinum describes how she traveled: she went home while on the horse.

Because the preposition á takes the dative case here.

Icelandic nouns change form depending on their grammatical role and on which preposition is used. The basic dictionary form is:

• hestur = horse

But after á in this sentence, the noun appears in the dative singular definite form:

• hestinum = the horse (dative)

So:

• hestur = horse, nominative
• hestinn = the horse, accusative
• hestinum = the horse, dative

Because the movement is toward home, not onto the horse.

This is a key Icelandic pattern:

• á + accusative often suggests motion onto something
• á + dative often suggests being on something

In Hún reið heim á hestinum, the phrase á hestinum describes her position while traveling: she was on the horse. The direction of travel is shown by heim, not by á.

So the sentence is not about getting onto the horse; it is about riding home while on it.

It is part of the Icelandic way of expressing the.

Unlike English, Icelandic usually attaches the definite article to the end of the noun instead of putting a separate word before it.

So:

• hestur = a horse / horse
• hestinum = the horse (in the dative singular)

In other words, -num is part of the definite dative ending here.

Yes. Hestinum is definite, so it means the horse — a specific horse, probably one already known from the context.

If you wanted to say on a horse or just on horseback without referring to a specific horse, you would normally say:

• á hesti

So the contrast is:

• á hesti = on a horse / on horseback
• á hestinum = on the horse

This word order is the most neutral one:

• Hún = subject
• reið = verb
• heim = direction
• á hestinum = prepositional phrase

So the sentence follows a very natural pattern: She rode home on the horse.

Icelandic word order is fairly flexible, but in main clauses the finite verb usually stays in second position. That means other word orders are possible for emphasis, but Hún reið heim á hestinum is the straightforward, neutral version.