As explained here on chabad.org:
The Rebbe, of righteous memory, stood for hours distributing dollars and blessings to thousands of people every Sunday, and on other occasions. The Rebbe’s intention was that the recipient should give the dollar to charity. In this way, explained the Rebbe, when two meet, it should benefit another.
Usually, instead of giving the actual dollar bill to charity, the recipient would keep it and give away another dollar in its place. I know it was a long time ago, but do you remember if you did that? If you do not remember, I would recommend giving a dollar to charity as soon as possible.
At the link it explains why this was specifically a crisp new dollar bill and why it was connected specifically to giving the blessings that many were asking for.
I saw on one of the YouTube videos, just a few days ago. (I don't have access to them, now, but there is one that focuses on a short "biography" of the Rebbe). At the beginning, the narrator syas that the Rebbe gave the dollars so that you should give it to charity.
If you can't locate the "bio" video, look at almost any other video. There's one about a woman coming to the rebbe to ask for a bracha and to get a Hebrew name. In it, the Rebbe says to the woman, "give this for charity".
I am assuming that in keeping with the Rebbe's general outgoing caring philosophy, he felt that it wasn't just sufficient to "bless" the recipient by helping him / her alone. The person had to "spread the blessing" by helping someone else, and donating the dollar to a charitable cause was a way to do this.
Of course, many people I know, including myself, kept the dollar as a memento. Then, again, many of them just used a different dollar or equivalent amount for charity, anyway.
BTW - I mentioned to my wife and kids, recently, that the Lubavitcher Rebbe zt"l was the only Rebbe I knew that gave you money when you visited a rebbe. Usually, it was / still is the other way around :-)
It was based on the saying of his father-in-law, the previous Lubavitcher Rebbe, that when two people come together, they should bring blessing to a third. The idea of the dollar was that it should be given to tzedakah.