I'm also interested in why specifically apples were used.
It's the ... Shaagas Aryeh, I think? Noda Bihuda maybe? -- who observes that some hosts throw challah across that table rather than pass it. He suggests that the sacrificial meat actually had to be thrown, rather than placed, onto the fire of the mizbeach, so perhaps this connects to this practice.
As for apples, my impression was that the rebbe is focused on kindness to his fellow Jews, so he should give something to everyone he encounters. An apple is convenient, affordable, and tasty.
(There's the story of the disabled individual who was seeking some additional assistance through a social services office of the Israeli government, they said they would send a social worker unannounced to his house to make sure he was legit. The fellow went to the rebbe for a bracha, and the rebbe gave him an ... I think in this story it was an orange. He had his whole family gather around and distributed pieces of the orange, as it was a big deal -- just then the social worker came by. (I thought the punchline would be, "this family is so poor, they can only afford one orange!" No, the social worker was impressed by the healthy family structure and put in her recommendation for financial assistance.)
Noheg B'am - page 87 in the second column at the bottom in the name of Sefer Mili L'Mordechai mentions this Minhag for Shavuos based on the Targum Sheni in Megilas Esther Chapter 3. Haman told Achashveirosh that in Sivan the Jews go to their synagogues and throw apples. He mentions this Minhag is mentioned by רבי דוד הסבעוני in the name of Rabeinu Maimon the father of the Rambam.
He goes on to mention in the name of Sefer Hazemanim that Jews are compared to apples where the fruit comes prior to the leaves, so to the Jews said Naase before Nishma.
Shefa Chaim 24 - page 103 discusses throwing apples on Simchas Torah. (My own thoughts - Perhaps there is a connection with finishing the Torah and Shavuos.)