How do I look through the issues of Puritan Reformed Journal which purchased today?

Not sure on which platform you're accessing your library or I'd be more specific. Find the library icon, click on it, type in the journal title in the search field, click on the issue you want to read and it should open. In the web app (app.logos.com), it looks like this:

Once you click on library, type the title in the field that says "Find":

In this example, I searched for NASB and found 10 titles in my library, I can click on any to open it and read:

From the Home tab, click on Contents to toggle on or off the table of content of the journal and go to a specific article. Or you can just use arrows and page down or scroll with your mouse:

On other platforms (desktop, mobile) look for a library button (same icon as the one shown above) and then it's pretty much the same except except that you can tap and drag on tactile screens like tablets or smartphones.