I think you misunderstood their question. Obviously immune response to common cold isn't effective against rona. But if this vaccine is effective against a wide range of coronaviruses, could it also be effective against the one that causes the common cold?
Normally when you get infected you get small number of viral particles in your system. Those then hijack your system to create more, and those new particles spread to other people through sneezing etc. In time your immune system learns how to recognise and kill the virus and eventually removes it from your system.
Now let's say you get vaccinated and you get that same small number of viral particles in your system. Your immune system will already know how to recognise and kill the virus, and immediately starts doing this. It doesn't multiply in your body, so it can't spread to others either.