How do zombies send mail?

I always wondered: how do infected computers send emails? I read all the stories of large botnets sending 1-2 billion letters a day. When I use my ISP's SMTP server and I try to send my newsletter to less than 100 people who block the ISP's SMTP server servers. I tried to do this with a few friends (with all the different internet providers), and everything was the same. But how can these hats create so much email when I can't even send 100?

Thanks in advance, Jory.