It's method extraction time!

But before that, I have to comment on your Player enum:

• str is a bad name, name would be better.
• Speaking of name, all enums have a name() method by default. You don't need your str variable, return name() instead
• Speaking of return name(), that's exactly what the default implementation of toString already does for enums.

And therefore, we've reduced your Player enum to:

public enum Player {
    X, O;
}

Ain't it lovely with enums? :)

Now, back to your getWinner() method:

This is just one version of doing it, it's not the optimal one but it should get you started. The idea here is that we add a bunch of Player objects to a list and then we check if those Player objects match to find a winner or not.

List<Player> players = new ArrayList<>();
for (int x = 0; x < fields.length; x++) {
    for (int y = 0; y < fields[x].length; y++) {
        players.add(fields[x][y]);
    }
    Player winner = findWinner(players);
    if (winner != null) 
        return winner;
}

Player findWinner(List<Player> players) {
    Player win = players.get(0);
    for (Player pl : players) {
        // it's fine that we loop through index 0 again,
        // even though we've already processed it. 
        // It's a fast operation and it won't change the result
        if (pl != win)
            return null;
    }
    return pl;
}

Please note that there are even more improvements possible for this getWinner method, I don't want to reveal all my secrets for now though ;) This is just one way of cleaning it up, which will reduce your code duplication a bit at least.

It's method extraction time!

But before that, I have to comment on your Player enum:

• str is a bad name, name would be better.
• Speaking of name, all enums have a name() method by default. You don't need your str variable, return name(); instead.
• Speaking of return name();, that's exactly what the default implementation of toString() already does for enums. Absolutely no need to override it.

And therefore, we've reduced your Player enum to:

public enum Player {
    X, O;
}

Ain't it lovely with enums? :)

Now, back to your getWinner() method:

You have a whole bunch of duplicated code there indeed. It would be handy if you could get a Collection of some kind (or an array), add some elements to it and check: Is there a winner given by these Player values?

This is just one version of doing it, it's not the optimal one but it should get you started. This code will add a bunch of Player objects to a list and then we check if those objects match to find if there's a winner.

List<Player> players = new ArrayList<>();
for (int x = 0; x < fields.length; x++) {
    for (int y = 0; y < fields[x].length; y++) {
        players.add(fields[x][y]);
    }
    Player winner = findWinner(players);
    if (winner != null) 
        return winner;
}

Player findWinner(List<Player> players) {
    Player win = players.get(0);
    for (Player pl : players) {
        // it's fine that we loop through index 0 again,
        // even though we've already processed it. 
        // It's a fast operation and it won't change the result
        if (pl != win)
            return null;
    }
    return pl;
}

Please note that there are even more improvements possible for the getWinner method, I don't want to reveal all my secrets for now though ;) And also, this is just one way of doing it, which will reduce your code duplication a bit at least. There are other possible approaches here as well.