Trust
Trust runs the world. Everything in business is about trust.
Do I trust this person enough to give them an interview? Do people trust you enough to hire you? Do people trust you enough to buy your products? Do they trust you enough to take your advice?
For a startup, landing a big client means you have their trust. Then you can leverage that trust as social proof. For a software engineer, getting a big company on your resume makes you more trustworthy.