Learn Romanian: El își dă seama că înțelesul mesajului e vag. - He realizes that the meaning of the message is vague.

The verbal locution for 'to realize' is a-și da seama. It functions with a fixed dative reflexive clitic, which literally translates as 'giving account to oneself'. Since the subject is the third-person el (he), the correct clitic is își (to himself).

Why does mesaj have the ending -ului in this sentence?

It is in the genitive case to express possession, translating to 'of the message'. Mesaj is a neuter noun, so its singular definite genitive ending is -ului. Notice that the noun being possessed, înțelesul, must also take a definite article (-ul) for this grammatical structure to work.

Could I use sensul instead of înțelesul here?

Yes, both sens and înțeles translate to 'meaning' and are highly interchangeable in this context. Înțeles is directly derived from the verb a înțelege (to understand), so it carries a slight nuance of 'how the message is understood' by the recipient.

Is it mandatory to use că in this sentence, or can it be omitted like 'that' in English?

In Romanian, the conjunction că (that) is strictly mandatory when connecting the main clause to the subordinate clause. You cannot drop it like you often do in English sentences such as 'he realizes the meaning is vague'.