Translation:Does anyone understand the meaning of this expression?
Both of the following should be acceptable:
את משמעות הביטוי הזה
את המשמעות של הביטוי הזה
You can't mix and match, though. The former is a smichut (סמיכות), which means that one word (משמעות) is "leaning" on the other word (הביטוי). The smichut is used in this case to indicate that something (the meaning in this case) belongs or is associated with something else (the expression). The smichut serves to indicate which meaning we are talking about (the one for this expression), so the meaning is already definite. You don't need and shouldn't add the definite article.
The latter is without smichut, so you need the word של to indicate the relationship between the words. You also need the definite article to indicate that this is a specific meaning you're talking about.