I'll mostly judge the actual parsing part, and not that much the boilerplate stuff around it
Why explode the file twice?
$element should be called $elements, and this loop won't really work as expected:
foreach ($element as $element)
There can be multiple selectors for one block:
h2, b {
...
Currently that counts as only one in your code.
What about comments?
/* This is for bold text*/
b {
gives /* This is for bold text*/ LINEBREAK b as name in your example. Certainly not right.
When you try to remove the element name from the key:values what happens if I put a space before the {? Actually, why not split $a_name[1]?
Counting to < count($styles) is enough (that also saves you the trimming); PHP indexes arrays from 0 on.
Then you never check for errors. If it doesn't have any { in it, it can't be a CSS file, no need to parse further. Or what if your $a_key_value has something but 2 elements? That's an error and should be noted somehow.
And so on for basically every step you do.
Read the specs when doing this. E.g. did you know about the many types of selectors (I only knew 4 of those and when I had written this parser I would totally have rejected E + F and similar)
For some security: An attacker could use your file_get_contents() call to open any file on your server. All he needs to do is set the file name to "/../../../../../etc/passwd".
You prepend some hash and some paths so your $this->newFileName becomes /var/www/example.com/HASH/../../../../etc/passwd. The ../ means go one directory up so your end up with just /etc/passwd.
And then you try to parse that as CSS. It's highly unlikely that it will contain valid CSS but as you currently don't try to stop the parsing, some information from it can be read.
