An agency already has the accounts you may not, such as a DSP account, a direct line to a publisher's sales team or a retail media platform, and it buys through them for you. You tell them your goal and your budget. They choose the ad spots, negotiate the rate, run the campaign and report back. You pay for their accounts and for the people who know how to use them, which is why an agency often makes sense when the buy is bigger than your team.
Buy direct when you know where you want to advertise and the publisher will sell to you, for example on a self-serve platform, in a newsletter with a rate card or on a site with an advertise page. Brief an agency when the buy covers more publishers than you can manage, when you need an account you don't have, or when the minimum spend is higher than a publisher will accept from a new advertiser. The ways to buy on each listing tell you which case you're in before you ask anyone.
These agencies buy advertising for their clients. The list is alphabetical, and nobody pays to be on it or to move up.