I noticed that too.
I believe that would be considered a voiceless I, not a "silent" I.
I looked up romanian phonology on Wikipedia, and there doesn't appear to be any voiceless vowel phonemes in Romanian, so my best guess is that voiceless I is an allophone of voiced I that appears only at the end of a word or something like that.
It would be great if someone who knows more about romanian linguistics could say a bit about this.
If I am not mistaken (and I very well might be) I at the end of a Romanian word usually palatalizes the previous consonant (similar to many Slavic languages). That means that it has the tongue kind of raised up toward the roof of the mouth. It is very slight difference and one that doesn't exist in English, which makes it very hard to hear.
That is what I picked up from another discussion. Palatilization used to be common in English with the vowel "u", but is disappearing. Think of the difference between "poor" and "pure" or between "moo" and "mew". In the second word of each pair the initial letter is palatilized. Other words that used to use the same sound are "news", "dew" and "tune" but now they are often replaced by nooz, doo and toon. I don't think there are any examples in English where palatilization occurs at the end of the word, making this doubly difficult to perceive.