Temperaturen stiger, når solen skinner.

Because -en is the usual singular definite ending for many Danish nouns of common gender.

So here:

temperatur = temperature
temperaturen = the temperature

sol = sun
solen = the sun

In English, the is a separate word. In Danish, it is often added to the end of the noun instead.

In Danish, a simple definite noun usually takes a suffix instead of a separate article.

So Danish prefers:

temperaturen = the temperature
solen = the sun

not a separate-word pattern like English.

A separate definite word such as den or det is usually used when there is an adjective before the noun, for example:

den varme sol = the warm sun
den høje temperatur = the high temperature

They are both in the present tense.

The infinitives are:

stige = to rise
skinne = to shine

The present tense is:

stiger = rises / is rising
skinner = shines / is shining

Danish present tense is the same for all persons, so it does not change like English I rise / he rises.

It can mean either, depending on context.

Danish usually uses the simple present where English might use either the simple present or the progressive. So:

solen skinner can mean
the sun shines or the sun is shining

In this sentence, it is best understood as a general statement: when the sun shines, the temperature rises.

Når usually means when in the sense of something general, repeated, or expected.

That fits this sentence well, because it expresses a general truth:

Temperaturen stiger, når solen skinner.

A useful comparison:

når = when / whenever, for general or repeated situations
da = when, for a specific event in the past
hvis = if

So this sentence uses når because it is not talking about one single past moment. It is talking about what generally happens.

Because når solen skinner is a subordinate clause.

In Danish, subordinate clauses normally keep the order subject + verb:

solen skinner

By contrast, main clauses in Danish follow the V2 rule, where the finite verb is in second position. But after a subordinating word like når, you do not use that main-clause inversion.

So:

Temperaturen stiger = main clause
når solen skinner = subordinate clause

Then the main clause must invert:

Når solen skinner, stiger temperaturen.

Notice that it becomes stiger temperaturen, not temperaturen stiger.

This is a very important Danish word-order rule: when something other than the subject comes first in a main clause, the verb still has to stay in second position.

The comma marks the beginning of the subordinate clause:

Temperaturen stiger, når solen skinner.

That helps show the structure clearly.

You should know, though, that Danish comma usage has allowed more than one system, so you may also see sentences without this comma in some writing. But in the sentence you were given, the comma is there to separate the main clause from the når clause.

Stiger is the most natural verb here.

For temperature, prices, numbers, and levels, Danish very often uses stige = rise / increase:

Temperaturen stiger = the temperature rises
Priserne stiger = prices are rising

bliver højere is understandable, but it sounds less natural for things like temperature. It is more literal, like becomes higher.

So a learner should treat temperaturen stiger as the normal idiomatic phrasing.

Not in this sentence.

Stige is usually intransitive: the thing itself goes up.

Temperaturen stiger = the temperature rises

Øge is usually transitive: someone or something increases something else.

For example:

Solen øger temperaturen = the sun increases the temperature

So in your sentence, temperaturen is the thing that goes up by itself, which is why stiger is the correct choice.