What you want to find in the layout folder inside your theme folder is a file called frontpage.php. This file defines how the frontpage looks.
In the file is a line that makes reference to login_info():
You can comment out the line as follows:
/* echo $OUTPUT->login_info(); anything between these markers is ignored */
(If you are using the Standard theme it doesn't have its own layout folder, the layout is defined in the Base theme).
Please be careful and make a backup copy of any file before you change it. Commenting is usually safe but if you are not good with php it's quite easy to make a mistake and then all you will see is a blank page.
Hope this helps.