The 'spiders' are web crawling bots that 'crawl' through (or browse) the web, usually for the purpose of indexing.
If you look at the language used, you'll often see the terms 'world wide web' and 'crawler bot'. That usually conjures up the image of 'spider' in a person's mind, as they tend to crawl on webs.
Web crawlers also visit many sites at the same time, so if you think about it, their 'legs' are positioned over many strands of the 'web', like a spider, so to speak.