What is, and more over is there a source for throwing of bread during Tashlich (other then the Kids song that says to throw our Aveiros away)? The reason for the the question is because on Rosh Hashanah I witnessed people doing it and realized it is problematic as feeding animals on yom tov is against Halacha. So if there is a source, does it allow for the feeding of animals on Yom Tov too?
There seems to be no known source for feeding the fish at Tashlich. As already noted, those who mention it then to be opposed to it.
But the fact is that it is mentioned - and its origins seem to be almost as old as the Minhag of Tashlich.
Regarding feeding the fish at Tashlich we could learn from the Aruch HaShulchan's opinion regarding feeding the birds on Shabbat Shira. He allows it for a variety of reasons.
But note, he says nothing about feeding fish when discussing Tashilch.
The OU page has this to say about feeding fish at Tashlich:
So feeding the fish is not necessarily forbidden - and those who do it have a Halachic to rely on - even if it has nothing to do with Tashlich, as far as anybody can ascertain.
In the seffer haminhagim of the Maharil, the source of tashlich quoted in the Ramma end of siman 583, he mentions in hilchos Rosh Hashana ois 9 when going to tashlich on Rosh Hashana to be carefull not to take food along so as not to feed the fish, which people would do for fun while they were there.
It has been proposed that this seemingly innocent act of pleasure eventually led to people associating tashlich with throwing bread and feeding the fish. This is not mentioned in any ancient sources. The Arizal would shake his clothes at tashlich. That was it. No emptying pockets of crumbs or the like. Shaking out clothes is an act associated with freshness and newness as we find in hilchos erev shabbos, if one doesn't own long clothing for shabbos he should shake out/down his weekday clothing. Stranger minhagim have gotten interpolated into this religion, but this one is definitely interesting.