Learn Hardcore Icelandic: Hún er spennt fyrir leiknum í kvöld.

Because adjectives agree with the subject in gender and number. The subject hún is feminine singular, and the feminine singular form of spenntur is spennt.

He: Hann er spenntur ...
She: Hún er spennt ...
Neuter singular: Það er spennt ...
Plurals: þeir eru spenntir, þær eru spenntar, þau eru spennt

What does fyrir do here, and why is leiknum in the dative?

The adjective spennt(ur) takes the preposition fyrir to introduce what you’re excited about: spennt(ur) fyrir e-u. In this meaning, fyrir governs the dative. Therefore, leiknum is the dative singular of leikur with the definite article (“the game”).

Noun: leikur (a game, masc.)
Dative singular (no article): leik
Dative singular (with article): leiknum

Why isn’t it leikinn after fyrir, like in Takk fyrir leikinn?

Because fyrir can take different cases with different meanings:

With thanks/benefit and some other meanings, it takes the accusative: Takk fyrir leikinn (“Thanks for the game”).
With the “excited about/for” pattern, spennt(ur) fyrir, it takes the dative: spennt fyrir leiknum.

What exactly is leiknum and how is it built from leikur?

Base noun: leikur (nom. sg.)
Dative sg. without article: leik
Definite article in the dative sg.: leiknum (“the game”) The article is a suffix, and in the dative singular masculine it appears as -num.

Could I say spennt um leikinn or spennt yfir leiknum?

spennt um is not idiomatic.
spennt yfir e-u does exist and often means “excited/ worked up about something (that has happened).” For upcoming events you look forward to, spennt fyrir e-u is the default.
- Good: Ég er spennt(ur) fyrir ferðinni.
- Also possible in the right context: Hún var spennt yfir fréttunum.

Can I drop fyrir and say Hún er spennt leiknum?

No. Many Icelandic adjectives require a specific preposition to link to their complement. Spennt(ur) needs fyrir (or sometimes yfir). Without the preposition, the sentence is ungrammatical.

Can í kvöld move to the front? What happens to the word order?

Yes: Í kvöld er hún spennt fyrir leiknum. Icelandic is verb‑second. When you front an adverbial like í kvöld, the finite verb er stays in second position, and the subject hún follows it.

Does í kvöld mean “tonight” or “this evening,” and how is it different from á kvöldin?

Í kvöld = tonight/this evening (the upcoming evening of today).
Á kvöldin = in the evenings (habitually, as a general time). Use í kvöld for “tonight,” not á kvöld.

How do I say “to the game” vs “at the game”?

Use á:

Motion to: á leikinn (accusative) → “to the game”
Location at: á leiknum (dative) → “at the game” Example: Hún er spennt að fara á leikinn í kvöld. (“She is excited to go to the game tonight.”)

How do I say “She’s looking forward to the game tonight”?

Use the verb hlakka til, which takes a genitive object:

Hún hlakkar til leiksins í kvöld. Here leiksins is the genitive singular definite of leikur.

What’s the difference between spennt(ur) and spennandi?

spennt(ur) = excited (how a person feels)
spennandi = exciting (what causes excitement) Examples:
Hún er spennt fyrir leiknum.
Leikurinn er spennandi.

How would the sentence change for “he” or for different kinds of “they”?

He: Hann er spenntur fyrir leiknum í kvöld.
They (all women): Þær eru spenntar fyrir leiknum í kvöld.
They (all men): Þeir eru spenntir fyrir leiknum í kvöld.
They (mixed/unspecified): Þau eru spennt fyrir leiknum í kvöld.

Why is there no article with í kvöld? Can I say í kvöldið?

Í kvöld is a fixed time expression and doesn’t take an article. Í kvöldið is not used in standard Icelandic.

Pronunciation tips for the whole sentence?

Approximate IPA: [huːn ɛr spɛnt fɪːrɪr ˈleiknʏm iː kvœlt]

Hún: [huːn] (“oo” as in “food”)
er: [ɛr]
spennt: [spɛnt] (final t clearly released)
fyrir: [fɪːrɪr] (rolled r; first vowel long)
leiknum: [ˈleiknʏm] (“ei” like English “ay”; u like “u” in “put”)
í: [iː] (“ee”)
kvöld: [kvœlt] (the ð is silent; ö like French “eu” in “peu”; final cluster sounds like “lt”)

Could I use the indefinite leik instead of leiknum?

Grammatically yes: Hún er spennt fyrir leik í kvöld would mean “excited for a game tonight.” But since í kvöld usually points to a specific, known game, speakers typically use the definite leiknum. Note that in other contexts fyrir leik can also mean “before a game,” so the definite form helps avoid ambiguity here.