The companies are seeking their "advices". When a CEO/CFO/CTO/C%O walks in his/her new role, s/he feels a strong need to get a second opinion about the business, to come up with new ideas or strategies to run the business better, to tap into new markets, alliances, etc.
No "everyone" can do the job. You have to be personal, likable, presentable, productive and smart. Being a Harvard graduate and on a US national team makes your clients want to talk to you for sure!