What does it mean when someone calls an article scholarly, academic, or peer reviewed?
Who writes these articles?
Experts in their fields. For literary analysis and criticism, that likely means professors and other scholars who research literature. They may, for example, devote their research to a small group of poets, or even one poet. They may be interested in a style or era as well.
Why do they write in these journals and not in magazines or newspapers or in books?
The most rigorous research happens in these journals. Publishing in these journals assures that your research will be read and cited by experts within your field of study. This research is the most up to date - once research is published here, it can be referenced for a general audience in newspapers or magazines, or it may be referenced in a more comprehensive work, such as a book. Having an article published in a journal is prestigious and may be a factor in getting and keeping one's job at a university. No, authors do not get paid by these publications.
How do I access these journals?
These journals are very expensive to subscribe to. This university provides access to many, and will borrow from other libraries on your behalf so you have virtually complete access to scholarship. You can search within the databases I recommend and Google Scholar (but use the link to GS that I provide since it links with our subscriptions).