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Why do VAT-registered businesses in the EU charge VAT to each other?
Not doing this would defeat the entire purpose of a VAT. The reason for a VAT rather than a simple sales tax is that it's harder to evade. Having a simple sales tax with the type of rates that VAT taxes typically are is unworkable because evasion is too easy. Imagine I'm a retailer. I buy products from a wholesaler and sell them to consumers. With a sales tax, if I don't charge the customer sales tax, the customer is happy and I don't care (assuming I don't get caught). And if I keep the sales tax but don't report the sale, I make a lot of money. Now, imagine a VAT. If I don't charge the customer the VAT, I lose money since I paid the VAT on the wholesale products. And if I don't report the sale, how do I claim my VAT refund?