console.log(artistsArray.name);

searchAlbums(){
    spotifyWebApi.searchAlbums(this.state.value)
    .then((response) => {
      return response.albums.items.map((t) => {
        return t.artists.map((artistsArray, index) => {
          this.render({
            render(){
              return (
                <div>
                <li key={artistsArray.name}>
                  {artistsArray.name}
                </li>
                </div>
              )
            }
          })
        });
      });
    });
  }

<button class="btn btn-small" type="button" onClick={() => this.searchAlbums()}>Search</button>