When you request a page such as http://localhost/index.php, think of it this way (very simplified):
1) The web server will look for a file with that name (index.php). The server is a program such as Apache's httpd.exe. This is important: the web server must get a request for a file that it can find. Okay, there are other options in other programming languages, but I'll ignore that complication for now.
2) If the server recognizes the file's extension as a script it can run (in this case index.php) it will run the script interpreter with that file as input (the interpreter is a program like php.exe or mod_php depending on the server and operating system).
3) The script interpreter will write a stream of data that is sent back to whatever requested it (usually your browser). At this point the interpreter is done running your file (index.php) so it exits or waits for another file to run.
3) The browser will look at the beginning of the data stream for a tag called the MIME TYPE and decide what to do with it (text, html, jpg, ...).
So where you're going wrong is having the script interpreter write a stream of data that contains php back to the browser. The browser thinks what you sent it is just regular text or html, it doesn't know anything about php. The web server never tried to run the php you read from the database.
So what does "eval" have to do with anything? While the interpreter is running, it can temporarily stop running the input file and run a script from somewhere else. So instead of running php code it found in index.php, the interpreter will run your php script from the database that you have in a variable.
Get it clear in your head that having the interpreter run the script is entirely different from having it write the php back to the browser.
Try this very simple example:
<?php
$foo = "Hello, world!\n";
eval('echo $foo;');
?>
If you want it to run the include file, try it like this:
test.php
<?php
echo "Hello ";
$f = "test_include.php";
include ($f);
?>
And the included file (make sure it has the <?php tags around it):
test_include.php
<?php
echo "World";
?>