Based on the context "insider" refers to being close to a source. So the website called "English Insider" has multiple meanings. One, the creator of the website is an English teacher, therefore has more knowledge of it than others. Two, the name implies that by visiting the website you can become more knowledgeable of English.
A more common way to use the word "insider" is in the context of criminal behavior. For example if someone knew that a company was going bankrupt and told a select group of major stockholders, before making a public announcement. That would be considered "insider trading."