Here's why and what your options are:
* No "self-programming" mode: Unlike some later model vehicles with straightforward key fob programming sequences, the 2003 Silverado doesn't have this feature built-in.
* Dealer or Locksmith: Your best bet is to visit a Chevrolet dealership or a qualified automotive locksmith. They have the necessary equipment (usually a scan tool capable of communicating with the vehicle's BCM - Body Control Module) to program the key fob correctly.
* Third-Party Scan Tools: Some advanced scan tools available for purchase online *might* be capable of programming the key fob, but you'll need one that explicitly supports this function for a 2003 Silverado and be comfortable with using the software. If you're not experienced with automotive diagnostic tools, this option is risky and could potentially cause more problems.
In short: Take it to a professional. Trying to program it yourself without the correct tools is unlikely to succeed and could lead to unnecessary frustration or damage.