An address is read before it is used. A bank opening an account, a client checking who they are contracting with, a supplier setting up terms — each of them looks at where a company says it is, and forms a view. A settled central address says the company intends to be around.
The second job is more practical. Business post has to arrive somewhere, be recognised as important, and reach the right person. That is easy when everyone sits in one office and rarely travels. It stops being easy the moment a founder is abroad, a team is distributed, or the trading premises change.
The third job is separation. An address used for business is visible: it appears on documents that are difficult to withdraw and on filings that stay on the public record. Keeping a home address out of that is a reasonable thing to want, and for many people it is the reason they take a business address at all.