Edit: I've updated the code and uploaded a gist that will work in the TS playground - the gist will compile with Typescript 3.0's strictest settings (at least all the strict settings available to the playground).

export interface Dictionary<K, V> {
  getKeys(): K[];
  getValues(): V[];
  get(key: K): V;
  put(key: K, val: V): void; // or boolean?
}


export class JSDictionary<K extends string, V> implements Dictionary<K, V> {

  private internalDict: Partial<Record<K, V>>;

  constructor() {
    this.internalDict = {};
  }

  public getKeys(): K[] {
    let keys: K[] = [];
    for(let key in this.internalDict) {
      keys.push(key);
    }

    return keys;
  }

  public getValues(): V[] {
    let vals: V[] = [];

    for(let key in this.internalDict) {
      vals.push(this.internalDict[key]);
    }

    return vals;
  }

  public get(key: K): V {
    return this.internalDict[key];
  }

  public put(key: K, val: V): void {
    this.internalDict[key] = val;
  }

}

type myKeys = 'FOX' | 'CAT' | 'DOG';

interface Animal {
  species: string;
  name: string;
  weight: number;
}

// A dictionary that hols one fox/cat/dog.
let myAnimalPen = new JSDictionary<myKeys, Animal>();

myAnimalPen.put('FOX', { name: 'Foxworth', species: 'Fox', weight: 40 });

// a dictionary that takes any string and maps it to a number
let idDict = new JSDictionary<string, number>();
idDict.put('somehas', 1204);
idDict.put('yeahaasd', 3306);

let yeaID = idDict.get('yeahaasd'); // yeaID is a number type
let myFox = myAnimalPen.get('FOX'); // myFox is an Animal type

Edit: I've updated the code and uploaded a gist that will work in the TS playground - the gist will compile with Typescript 3.1.

export interface Dictionary<K, V> {
  getKeys(): K[];
  getValues(): V[];
  get(key: K): V | null; // the key might not exist
  put(key: K, val: V): void; // or boolean?
}


export class JSDictionary<K extends string, V> implements Dictionary<K, V> {

  private internalDict: { [key in K]?: V };

  constructor() {
    this.internalDict = {};
  }

  public getKeys() {
    let keys: K[] = [];
    for(let key in this.internalDict) {
      keys.push(key);
    }

    return keys;
  }

  // Type predicate to ensure v exists
  private exists(v: V | undefined): v is V {
    return v != null && typeof v !== "undefined";
  }

  public getValues() {
    let vals: V[] = [];

    for(let key in this.internalDict) {
      let v = this.internalDict[key];
      if(this.exists(v)) {
        vals.push(v);
      }
    }

    return vals;
  }

  public get(key: K) {
    let v = this.internalDict[key];
    return this.exists(v)
      ? v
      : null;
  }

  public put(key: K, val: V): void {
    this.internalDict[key] = val;
  }

}

type myKeys = 'FOX' | 'CAT' | 'DOG';

interface Animal {
  species: string;
  name: string;
  weight: number;
}

// A dictionary that hols one fox/cat/dog.
let myAnimalPen = new JSDictionary<myKeys, Animal>();

myAnimalPen.put('FOX', { name: 'Foxworth', species: 'Fox', weight: 40 });

// a dictionary that takes any string and maps it to a number
let idDict = new JSDictionary<string, number>();
idDict.put('somehas', 1204);
idDict.put('yeahaasd', 3306);

let yeaID = idDict.get('yeahaasd'); // yeaID is a number | null type
let myFox = myAnimalPen.get('FOX'); // myFox is an Animal | null type

The primary benefit of Typescript is to provide types for Javascript. And since Typescript provides Generics - I don't think your implementation of a Dictionary provides any benefit over a plain javascript object.

Edit: I've updated the code and uploaded a gist that will work in the TS playground - the gist will compile with Typescript 3.0's strictest settings (at least all the strict settings available to the playground).

The primary benefit of Typescript is to provide types for Javascript. And since Typescript provides Generics - I don't think your implementation of a Dictionary provides any benefit over a plain javascript object.