Það er skemmtilegra að lesa í skóginum en heima.

Það er ... is a very common impersonal way to start sentences in Icelandic, similar to English It is ... in It’s fun to...
Here það doesn’t point to a specific thing; it’s a “dummy” subject used to introduce an evaluation of an action: It is more enjoyable to read...

Because skemmtilegra is the comparative form of skemmtilegur (fun / enjoyable).

• Base form: skemmtilegur (m.), skemmtileg (f.), skemmtilegt (n.)
• Comparative: skemmtilegri (m/f), skemmtilegra (n)

In this Það er ... structure, the adjective is often used in the neuter singular form, so you get skemmtilegra.

The -ra marks the neuter singular comparative form for many adjectives. It corresponds to English more ... (or -er) in comparisons.
So skemmtilegra = more enjoyable / more fun.

að is the normal marker used before an infinitive in Icelandic in many constructions—similar to English to.
So að lesa = to read.

(You’ll also see infinitives without að after some verbs, but after an evaluation like Það er skemmtilegra ..., að is expected.)

í skóginum means in the forest, and it uses:

• the preposition í (in)
• the noun skógur (forest) in the dative case because í
  • location/static position takes dative

skóginum = the forest in dative singular definite (roughly: in + the forest).

With í (and also á), the case depends on meaning:

• Dative = location (no movement): lesa í skóginum (read in the forest)
• Accusative = motion/direction: fara í skóginn (go into the forest)

Here you’re already there reading, so it’s dative: í skóginum.

en is the standard word for than in comparisons:

• skemmtilegra ... en ... = more enjoyable ... than ...

You may also encounter heldur en in some contexts (often for emphasis or contrast), but plain en is the default and very common.

heima is an adverb meaning at home. It’s a fixed form and very common in everyday Icelandic.
So ... en heima is literally ... than at home.

You can express similar ideas with other phrases, but heima is the most natural equivalent of English at home in sentences like this.

Yes. Both are possible:

• Það er skemmtilegra að lesa í skóginum en heima. (very common, “It’s more fun to...”)
• Að lesa í skóginum er skemmtilegra en heima. (also correct; puts more focus on reading in the forest as the topic)

The Það er ... version often sounds more conversational and neutral.