When setting up a DD, you can give a european company your IBAN. They will be able to make withdrawal requests, directly to your bank account. Your bank will usually give you full control on what you can do with those. For example, my bank Bunq (https://bunq.me/) allows me to accept once, or automatically accept any debit request up to whatever amount I choose per period from that same merchant.

Direct Debits are free. You bypass credit card fees (which in Europe are much lower than in the US in the first place but still percentage-based). They are also usually real-time (though slower than the credit card network, and depending on banks and fraud checks there can be latency of up to a couple of days).