Search operators come in handy in SEO for many reasons:

Whenever optimizing a new site from a new vertical they are the easiest way to find the relevant websites where your site should be promoted on.

[keyword] blog list
[keyword] news
[keyword] forum
[keyword] “guest post”
[keyword] “roundup”
[keyword] resources”

The same when having to find relevant sites where you could promote the website through guest posts.

keyword “guest post”
keyword inurl:category/guest
keyword “guest article”
keyword “write for us”

And so on.

Advanced search operators let you refine the result so much that you can find exact pages you are looking for.

Say you’d like to find keyword relevant articles only from .com that will allow you to leave a comment, here’s what the formula would look like:

“add comment” intitle:keyword

Whatever you’d like to find, the limit is only your imagination.

To familiarise yourself with the operators you can start using Google Advanced Search & studying the options they offer, and go through this page about Advanced Search Operators.