Questions & Answers about Hún leitar að lyklunum í handtöskunni sinni.
Because the verb leita usually takes the pattern leita að + dative when it means look for / search for.
So in this sentence:
- leitar = looks for / is looking for
- að lyklunum = for the keys
Here, að is not the English-style to. It is just part of the construction that this verb uses.
A useful thing to memorize is:
- leita að einhverju = look for something
- leita að einhverjum = look for someone
So the case after að here is important: it must be dative.