When to add type annotations

TL;DR

A good rule of a thumb is to not add types. Unless you have to, because TS didn't infer the type you wanted.

Type Inferance

TypeScript is quite smart when it comes to type inference. For example:

// the `number` annotation is redundant; it doesn't really do anything
let one: number = 1;
// you might as well omit it and save yourself a few keystrokes:
let one = 1;

When to add types to variables

But TypeScript has it limits. Here's when you have to be more specific:

let primeNumbers = [];
primeNumbers.push(2);
primeNumbers.push(3);
primeNumbers.push(5);
// oops, your probably didn't mean to do that
primeNumbers.push("seven");

TypeScript infers primeNumbers to be of type any[]. So even though the author of the code probably meant it to be of type number[], TS still allows a string to be pushed to the array.
The reason TS wasn't able to infer the type, because TS had to little information about the variable - if the array is empty, how am I supposed to know what's in it? That wouldn't have been an issue, if you init an array with some valuese:

let primeNumbers = [2];
primeNumbers.push(3);
primeNumbers.push(5);
primeNumbers.push(7);
// this won't compile
primeNumbers.push("eleven");

If you have to start with emptyarray, why you have to explicitly add the type:

let primeNumbers: number[] = [];
primeNumbers.push(2);
primeNumbers.push(3);
primeNumbers.push(5);
// this won't compile
primeNumbers.push("seven");