Learn Russian: Я понимаю проблему.

The dictionary form of the word is проблема. Because it is the direct object of the verb (the thing being understood), it must be in the accusative case. For feminine nouns ending in -а, the ending changes to -у.

I thought inanimate objects don't change in the accusative case?

That is only true for masculine and neuter nouns! Feminine nouns ending in -а always change to -у in the accusative case, regardless of whether they are animate (alive) or inanimate (not alive).

Why does the verb понимаю end in -ю?

This is the conjugated form for Я (I). The dictionary form is понимать (to understand). For most verbs like this, you drop the -ть and add -ю to show that 'I' am doing the action.

How do we know it means 'the problem' and not 'a problem'?

Russian does not have articles like 'a' or 'the'. The word проблему can mean either one, so you have to figure it out from the context of the conversation.