Shaking like a peach tree just simply means that you're shaking pretty bad.
You could be shaking due to being cold, being scared, nervous etc.
In the wind a peach tree could shake and someone saying you're shaking like a peach tree is just saying that you're shaking uncontrollably.
My dad used to always tell me I was shaking like a peach tree when I was nervous about something or came in out of the cold and was shaking to warm up.
Our bodies when cold shake to help our bodies warm up.
As we shake our bodies create heat which in turn warms us up a little bit.