I can give you some examples how some bad web hosting services can affect your SEO (and the performance of the website/websites at all):
1. Site is down for hours or even for days. Obviously, it will affect your SEO.
2. Web hosting uses some strict protection and it blocks the SE spiders to crawl your content.
3. Web hosting uses some strict protection and you even can't post new content.
Please, feel free to add more.
I don't want to talk about which web hosting is the best, because I don't use many and I can't compare them. Lack of information.