Before Google, search engines often failed to return any relevant results for keywords. Search engines at that time only used a site's on-page content to figure out its relevance for different keywords. This changed with Google, which came up with a patented method of trying to find the relevance of a site from the number and types of links that pointed to it. That directly impacted the importance of anchor text in SEO because every link comes with an anchor text.
The anchor text is the textual part of any link, the part that is displayed to the user. Anchor text normally informs a user about where the link is taking them, or what the link is about. However, it is possible to have an anchor text say "carrots" even when the link is taking the viewer to a site about radishes. But, as is always the case, don't just do what is possible. Not only is this misleading for the user, Google does not approve of anchor text that is different from the actual site that the link points too.
Another recent change is that now Google checks that the anchor text on a page is relevant to the rest of the content on that page. If you have a webpage that talks about gardening, and all your anchor text are about payday loans, Google will know that something unusual is going on, and could penalize the site. This new change came with the latest Panda and Penguin updates.
Google also looks down on the aggressive use of keywords as anchor text to the exclusion of anything else. A good anchor text should tell the viewer more about the link than just the keyword. For example, if the link is about carrot seeds, it is better to have anchor text that is different than just "carrot seeds." A more descriptive anchor text would be something like "check out these carrot seeds." Other generic anchor texts to throw into the mix include "more info", "click here" and "check this out." Google understands that this natural anchor text is written for the ease of the reader, and the reader is Google's main focus.
Anchor text is still an important part of a good SEO strategy, but it should be used conservatively. Use a balanced and natural combination of keyword anchor text, brand anchor text and generic anchor text.