How are businesses able to use experimental aircraft? I saw businesses like this using military jets to sell rides to civilians. Since warbirds are considered experimental aircraft, does the FAA issue special certs to fly them commercially? 

According to 14 CFR § 91.319, 


  (a) No person may operate an aircraft that has an experimental certificate
  
  
    (1) For other than the purpose for which the certificate was issued; or
    
    (2) Carrying persons or property for compensation or hire.
Waivers available for the purpose of instruction. EAA can help with this. 
You will not be able to instruct anyone who does not already have at least private pilot license. 
You can of course use the airplane as a vehicle personal transport. If you a high value individual traveling to a client and get paid for travel time, paying for the use of your vehicle could actually cost the customer less.