Líkamsræktin styrkir líkamann.

The ending -in is the definite article attached to the noun, so it means the.

• líkamsrækt = exercise / physical training / fitness
• líkamsræktin = the exercise / the physical training

In Icelandic, the is very often added to the end of the noun instead of being a separate word like in English.

Because líkamann is:

• the noun líkami = body
• in the accusative singular
• with the definite article attached

So:

• líkami = a body
• líkamann = the body (as a direct object)

In this sentence, the body is what gets strengthened, so it appears in the accusative case.

Yes, they are related.

Both come from the noun líkami = body.

The word líkamsrækt is a compound word:

• líkams- = a form of líkami
• rækt = roughly cultivation / training / development

So líkamsrækt literally means something like body-training, which is why it is used for exercise or physical fitness.

Then later in the sentence you also get líkamann, which is the actual word the body as the object.

So the sentence contains:

• a compound built from body
• and then the body itself

Because Icelandic compounds often use a linking form based on the genitive.

The noun is:

• líkami = body

Its genitive singular is:

• líkama or in older/compound-forming usage líkams as the linking element in compounds

So in compounds you often see:

• líkamsrækt = body-training
• líkamshiti = body temperature
• líkamshluti = body part

You do not usually need to translate this -s separately; it is just part of how compounds are formed.

styrkir is the present tense, 3rd person singular form of the verb styrkja, which means to strengthen.

So:

• að styrkja = to strengthen
• ég styrki = I strengthen
• þú styrkir = you strengthen
• hann/hún/það styrkir = he/she/it strengthens

In this sentence, the subject is líkamsræktin, which is singular, so the verb is also singular:

• Líkamsræktin styrkir ... = Exercise strengthens ...

Because styrkja normally takes a direct object, and direct objects are commonly in the accusative in Icelandic.

Here the structure is:

• líkamsræktin = the subject
• styrkir = the verb
• líkamann = the direct object

So the sentence is literally:

• The exercise strengthens the body

Since the body is the thing being acted on, it is in the accusative.

The dictionary forms are:

• líkamsrækt = exercise / physical training / fitness
• styrkja = to strengthen
• líkami = body

The sentence uses inflected forms of these:

• líkamsræktin = definite singular form of líkamsrækt
• styrkir = present 3rd person singular of styrkja
• líkamann = accusative singular definite form of líkami

It is singular.

Even though English might sometimes use words like exercise in a broad or uncountable way, Icelandic can still treat líkamsrækt as a singular noun meaning:

• exercise
• physical training
• fitness activity

So líkamsræktin styrkir líkamann means exercise strengthens the body, with exercise understood as a general activity.

Because Icelandic usually attaches the definite article to the end of the noun.

So instead of saying a separate word like English the, Icelandic often does this:

• líkamsrækt → líkamsræktin = the exercise
• líkami → líkamann = the body (in this case-form)

This is one of the most noticeable differences from English.

Yes, in this sentence it is basically the same as standard English word order:

• Subject: Líkamsræktin
• Verb: styrkir
• Object: líkamann

So it follows SVO order:

• Exercise strengthens the body

That said, Icelandic can sometimes move words around more freely than English because case endings make grammatical roles clearer.

A rough English-friendly guide would be:

• Líkamsræktin ≈ LEE-kams-rehk-tin
• styrkir ≈ STIR-kir
• líkamann ≈ LEE-ka-man

A few useful notes:

• í is a long ee sound.
• y in modern Icelandic is pronounced like i, so styrkir has an i-like vowel.
• r is rolled or tapped.
• kk and k can sound more strongly aspirated than in English.

This is only an approximation, but it is enough to get started.

Yes. That would mean something like:

• Exercise strengthens the body
• Physical training strengthens the body

without specifically saying the exercise.

Compare:

• Líkamsrækt styrkir líkamann. = Exercise strengthens the body.
• Líkamsræktin styrkir líkamann. = The exercise strengthens the body.

In many contexts, the version without the article may sound more natural if you mean exercise in general.

No, but it is a very common and natural word for physical exercise / fitness training.

Depending on context, Icelandic might also use:

• æfing = exercise / practice / workout
• hreyfing = movement / exercise
• að æfa sig = to practice / to work out

But líkamsrækt is a very standard word when talking about exercise as a health or fitness activity.

Because styrkir is the simple present form.

In Icelandic:

• styrkir = strengthens / does strengthen
• a past form would look different, for example styrkti = strengthened

Icelandic does not use tense forms in exactly the same way English does, but here styrkir is clearly a present-tense form.