Rótin er stutt núna, en hún verður lengri þegar plantan fær meiri sól.

Because Icelandic usually puts the definite article onto the end of the noun.

• rót = root
• rótin = the root
• planta = plant
• plantan = the plant

The exact ending changes depending on the noun’s gender and shape, so you do not always get the same article ending on every noun.

These are adjective forms agreeing with rótin.

• The dictionary form is stuttur = short.
• With a feminine singular subject like rót, the form here is stutt.
• lengri is the comparative form, meaning longer.

So:

• Rótin er stutt = the root is short
• hún verður lengri = it becomes longer

Also, after vera and verða, Icelandic uses a predicate adjective, and that adjective agrees with the subject.

er describes the current state, while verður describes a change.

• er = is
• verður = becomes / will become

So the sentence contrasts:

• how the root is now
• how it will change later

This is very common in Icelandic: verða + adjective often means become or get.

Because rót is a feminine noun, and Icelandic pronouns follow grammatical gender.

So a feminine noun is referred to with hún.

One small complication here: planta is also feminine, so purely from grammar, hún could match either noun. In this sentence, context tells you it refers to rótin, because the thing that becomes longer is the root.

Yes, þegar can mean both when and already in Icelandic.

Here it means when, because it introduces a full clause:

• þegar plantan fær meiri sól

When þegar is followed by a clause like that, it is functioning as a conjunction meaning when.
When it appears as an adverb by itself, it can mean already.

Because Icelandic often uses the present tense in time clauses referring to the future.

That is very similar to English:

• when the plant gets more sun not
• when the plant will get more sun

So þegar plantan fær meiri sól is the normal way to say when the plant gets more sun.

The base form is fá, which means get or receive.

fær is the 3rd person singular present form:

• ég fæ = I get
• þú færð = you get
• hann/hún/það fær = he/she/it gets

It is an irregular verb, so the stem changes.

Because meiri is the form used here with sól.

A useful way to think about it:

• meiri = more, used with nouns like this
• meira is typically the neuter singular form or an adverbial form

Also, meiri is the comparative of mikill, so it literally works like more.

Because sól here means sun/sunlight in a general, amount-based sense.

The phrase is about receiving more sunlight, not about a specific sun being definite. So Icelandic naturally says:

• meiri sól = more sun / more sunlight

This is similar to English, where we normally say more sun, not more the sun.

Because þegar introduces a subordinate clause.

In the main clauses, Icelandic often shows normal main-clause verb behavior, but in subordinate clauses like this one, the order is typically:

• conjunction + subject + verb + rest

So:

• þegar = when
• plantan = the plant
• fær = gets
• meiri sól = more sun

That makes þegar plantan fær meiri sól the expected and natural order.