Learn Icelandic: Hann æfir oft til að auka sjálfstraustið. - He practices often to increase his self-confidence.

Both mean 'in order to' or just 'to' when expressing a purpose. til að is simply a shorter, slightly more casual version of til þess að. Because it comes before the infinitive verb auka, you can use either one here interchangeably.

Why is auka in the infinitive form?

The purpose phrase til að must always be followed by a verb in its infinitive (basic dictionary) form. Here, it connects his practice directly to the action of increasing (auka).

Why does the English translation say 'his self-confidence' when there is no word for 'his' in the Icelandic sentence?

Icelandic often uses the definite article instead of a possessive pronoun for things deeply connected to a person, like body parts or personal traits. The suffix -ið makes sjálfstraust definite (literally 'the self-confidence'), but because it is obvious whose confidence is being increased, English naturally translates it as 'his'.

Does the verb auka force a specific case on sjálfstraustið?

Yes, auka (to increase) takes a direct object in the accusative case. sjálfstraust is a neuter noun, and for all neuter nouns, the accusative form is completely identical to the nominative form, which is why the word doesn't look like it has changed.