Learn Croatian: Trebamo čaše i tanjure. - We need glasses and plates.

Because it is the direct object of the verb (it is what we need), so it must be in the accusative case. While masculine plurals normally end in -i when they are the subject, they change to -e in the accusative plural.

I remember that inanimate masculine nouns don't change in the accusative. Why did tanjur change here?

That is a great rule to remember, but it only applies to the singular! In the singular, a lifeless object like a plate (tanjur) doesn't change its ending. But in the plural, all masculine nouns take the -e ending in the accusative, whether they are living or not.

Is čaše also in the accusative case here?

Yes! Since we need the glasses, čaše is the direct object too. For feminine nouns that end in -a (like čaša), the plural ending is -e. Luckily, feminine nouns look exactly the same in both the nominative plural (the subject form) and the accusative plural.

Why is there no word for "we" in this sentence?

In Croatian, you can usually drop pronouns like mi (we). The -mo ending on the verb trebamo already tells us exactly who is doing the needing. You would only add mi if you wanted to emphasize that we need them, as opposed to someone else.