Learn Finnish: Etsin uutta asuntoa tästä kerrostalosta. - I am searching for a new apartment in this apartment building.

This is because of the verb etsiä (to search). In Finnish, you don't search 'in' a place; you search 'out of' it to extract what you are looking for. Therefore, etsiä requires the location to be in the elative (-sta/-stä) or ablative (-lta/-ltä) case.

Why is the object uutta asuntoa in the partitive case rather than the accusative?

Etsiä is a partitive verb. Because the action of searching is ongoing and the result is uncertain (you haven't found the apartment yet), the object being looked for must take the partitive case.

Why does the word for 'this' change to tästä?

In Finnish, demonstrative pronouns like tämä (this) must agree with the noun they modify. Since the building is in the elative case (kerrostalosta), the pronoun must also take the elative ending, changing tämä to tästä.

How does the compound word kerrostalo break down?

It is made up of kerros (floor, layer, or story) and talo (house or building). Together, they literally mean a 'multi-story building', which is the standard Finnish word for an apartment building or block of flats.