$a++ is the same as saying $a+1
The ++ means increment, -- means decrement.
Yes, all PHP variables and object start with the $ sign.
$cell_cnt = 0; means that I am setting variable cell_cnt (which I called it that for easy recalling) to 0;
if($cell_cnt % 3 == 0){ echo '<tr>'; } means that: if $cell_cnt modulus 3 equals 0, print <tr>. Modulus is a math term meaning, get the remainder. So, if the remainder of $cell_cnt is equal to 0, put a <tr>
count($row_stories); means the same as, return the number of $row_stories.
So, if($cell_cnt == count($row_stories)), means if $cell_cnt is equal to the number of entries, end the final row.
Hope that helps.