Hún vill tjalda nálægt fossinum, en ég vil frekar vera á rólegu tjaldstæði.

They are different present-tense forms of the verb vilja (to want).

• ég vil = I want
• þú vilt = you want
• hún/hann vill = she/he wants

So vill is the 3rd person singular form, while vil is the 1st person singular form.

Because vilja is followed by a bare infinitive in Icelandic.

So:

• hún vill tjalda = she wants to camp
• ég vil vera = I want to be / stay

English uses to, but Icelandic does not use að here.

Tjalda means to camp in a tent or to pitch a tent. It is more specific than broad English camp, because it is connected to tents.

So hún vill tjalda suggests tent-camping, not just staying somewhere outdoors in a general sense.

In Icelandic, the definite article is usually attached to the end of the noun.

• foss = waterfall
• fossinn = the waterfall
• fossinum = the waterfall in the dative case

So where English has a separate word the, Icelandic often uses an ending instead.

Because nálægt (near) takes the dative case.

The noun foss therefore has to appear in its dative singular definite form:

• base noun: foss
• dative definite: fossinum

So nálægt fossinum means near the waterfall, with fossinum there because of the grammar after nálægt.

Yes, in this sentence it functions like a preposition meaning near.

It introduces a noun phrase and requires the dative, which is why you get fossinum. Learners often just remember it as:

• nálægt + dative

Here frekar means rather, preferably, or instead.

So:

• ég vil frekar vera... = I would rather be...

It shows preference between two options: camping near the waterfall versus being at a quiet campsite.

Yes, vera literally means to be. But when talking about location, it often naturally translates as be or stay in English.

So:

• vera á rólegu tjaldstæði literally = be at a quiet campsite
• natural English = stay at a quiet campsite

This is very common in Icelandic.

Because á takes the dative when it describes location, and vera describes being in a place, not moving to it.

So the phrase is dative:

• á rólegu tjaldstæði = at a quiet campsite

The adjective has to agree with the noun:

• tjaldstæði is neuter singular
• dative singular neuter of rólegur is rólegu

A useful contrast is:

• vera á rólegu tjaldstæði = be at a quiet campsite
• fara á rólegt tjaldstæði = go to a quiet campsite

Location usually gives dative; motion toward usually gives accusative.

Because the sentence means a quiet campsite, not the quiet campsite.

• tjaldstæði = a campsite / campsite
• tjaldstæðið = the campsite

So á rólegu tjaldstæði is indefinite. If you wanted at the quiet campsite, you would say:

• á rólega tjaldstæðinu

Here en means but. It links two contrasting ideas:

• Hún vill... = She wants...
• en ég vil... = but I want...

One extra point: en can also mean than after a comparative, so context matters.

A little, yes, but the given order is the most natural and safest for learners.

• Hún vill tjalda nálægt fossinum is a normal clause.
• ég vil frekar vera á rólegu tjaldstæði is also a normal clause.

Icelandic does allow movement for emphasis, but the finite verb usually stays early in the clause. So for a learner, the sentence as given is a very good model to follow.