食べていいよ (たべていいよ) - you may eat…
This is one of the first uses of the te (て) form that we learn as Japanese language students. When you come to Japan you will notice that everyone seems to often add an extra て to these types of sentences:
I am assuming the extra て comes from いて but in a shortened form, but this might not be correct.
In any case, after asking some questions it seems that though the two sentences above technically have the same meaning, the second implies that ‘I am not going to eat yet…’, that there may be some length of time involved, ‘go ahead and eat,’ and so on.
In the same way, while 持って (もって) means ‘please hold (this)’, 持ってて implies ‘please hold (this)… for a while’.