cannot assign to `value`, which is behind a `&` reference

Question

1. I don't understand why let mut first = in this situation is not good enough for the rust compiler to allow first to be mutable and change first.value
2. Is this some strange behavior because its contained inside a Box?
3. What does mut on the right side do that the left mut doesn't?

rustc --explain E0594 doesn't really explain much in this case.

#![allow(unused)]
fn main() {
    let mut all_my_data = Vec::new();
    
    all_my_data.push(Box::new(
        Data::new("Hello World!"),
        ),
    );
    all_my_data.push(Box::new(
        Data::new("This is the end of the World!"),
        ),
    );
    all_my_data.push(Box::new(
        Data::new("Goodbye!"),
        ),
    );
    
    let mut first = &all_my_data[0]; // This errors out
    //let mut first = &mut all_my_data[0]; // This works
    first.value = String::from("Hello New World!");
    println!("First: {:?}", first);
    let second = &all_my_data[1];
    println!("Second: {:?}", second);
    let third = &all_my_data[2];
    println!("Third: {:?}", third);
}

#[derive(Debug)]
struct Data {
    value: String,
}

impl Data {
    pub fn new(value: &str) -> Data {
        Data {
            value: value.to_string()
        }
    }
}

error[E0594]: cannot assign to `first.value`, which is behind a `&` reference
  --> src/main.rs:20:5
   |
18 |     let mut first = &all_my_data[0]; // This errors out
   |                     --------------- help: consider changing this to be a mutable reference: `&mut all_my_data[0]`
19 |     //let mut first = &mut all_my_data[0];
20 |     first.value = String::from("Hello New World!");
   |     ^^^^^^^^^^^ `first` is a `&` reference, so the data it refers to cannot be written

For more information about this error, try `rustc --explain E0594`.
error: could not compile `playground` due to previous error

Playground