Learn French: Combien de boîtes de tomates est-ce que vous achetez ? - How many boxes of tomatoes are you buying?

When using an expression of quantity like combien (how many/how much), it is always followed by de (or d' before a vowel). You never use des, even when the next word is plural.

Why do we say boîtes de tomates and not boîtes des tomates?

Just like with combien de, when you describe a container and its contents (like a box of tomatoes, or a cup of tea), you connect them with just de instead of a plural article.

What is the role of est-ce que in this sentence?

est-ce que is a common building block used to form a question. This sentence uses a very standard question structure: Question phrase (Combien de boîtes de tomates) + est-ce que + subject and verb (vous achetez).

Why does the English translation say "are you buying" when the French word is just achetez?

French doesn't have a separate "-ing" tense for actions happening right now. The present tense vous achetez can translate to "you buy", "you are buying", or "do you buy", depending on what makes the most sense in English.