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I moved this code in D15146, I'll be aware when landing.

Landed D15146, this type was moved and needs to be updated

Indeed it was wrong, changed it to something better.
Before we had
declare type Bind<D> = <A, R>((...A) => R) => (...A) => $Call<D, R>;

$Call<D, R> = the result of calling function D with argument R.
So the closes equivalent would be ReturnType<D>:
declare type Bind<D> = <A, R>((...A) => R) => (...A) => ReturnType<D>;

ObjMap<T, F> takes an object type T, and a function type F, and returns the object type obtained by mapping the type of each value in the object with the provided function type F. 
$ObjMap<DP, Bind<D>> =
take DP and return object obtained by calling Bind<D>(values of DP).
So for now we have:
{[key in keyof DP]: Bind<D>(DP[key])}
But we can’t call Bind<D> like that, because we can’t call types. Let's use ReturnType<>:
{[key in keyof DP]: ReturnType<Bind<D>>}
But as you look at the definition of Bind<D> you’ll see it’s a function that takes (...A) => R and returns (...A) => ReturnType<D>.
So we use infer to check what’s A and return a function based on that in MapDispatch

Changed any to something nicer I think. If T is something like this:

{ 
   key1: Promise<string>,
   key2: number
}

then PromiseAllResult should be:

{
   key1: string,
   key2: number
}

so in short PromiseAllResult extracts whatever is in the promises.