How to post data to get correct result?

I am trying to do some post.

Uses
  IdBaseComponent, IdTCPConnection, IdTCPClient, IdHTTP, IdComponent, StrUtils,
  IdSSLOpenSSL;

function PostData(const AURL: string; AParamList: TStringList): string;
var
  _idHTTP: TIdHTTP;
  _ResultStream: TStringStream;
  _IdSSLIOHandler: TIdSSLIOHandlerSocketOpenSSL;
begin
  _ResultStream := TStringStream.Create('', TEncoding.UTF8);
  _idHTTP := TIdHTTP.Create(nil);
  _IdSSLIOHandler := TIdSSLIOHandlerSocketOpenSSL.Create(nil);
  try
    _idHTTP.IOHandler := _IdSSLIOHandler;
    _idHTTP.Post(AURL, AParamList, _ResultStream);
    Result := _ResultStream.DataString;
  finally
    FreeAndNil(_ResultStream);
    FreeAndNil(_idHTTP);
    FreeAndNil(_IdSSLIOHandler);
  end;
end;

procedure TForm1.BitBtn1Click(Sender: TObject);
var
  _ParamList: TStringList;
  _Reply: string;
begin
  _ParamList := TStringList.Create;
  try
    _ParamList.Add('id=276683492511635');
    _ParamList.Add('pin=4580');
    _Reply := PostData('https://eturundus.eu/samsung-pin-lt/_ajax/pin',
      _ParamList);
    ShowMessage(_Reply);
  finally
    FreeAndNil(_ParamList);
  end;
end;

I should get { "status":"lose","tries":2 } or { "status":"over" }, etc, but all I am getting is an empty string in a message box. What I am missing?

Edit: I have added

_idHTTP.Request.UserAgent := 'Mozilla/5.0 (Windows NT 6.1; WOW64; rv:29.0) Gecko/20100101 Firefox/29.0';
_idHTTP.Request.ContentType := 'application/json, text/javascript, */*; q=0.01';

But still getting empty string.

There is what Tamper Data also provides me:

<?xml version="1.0"?>
   <!-- Export of tamperdata transacions -->
  <tdRequests>
    <tdRequest uri="https%3A//eturundus.eu/samsung-pin-lt/_ajax/pin%3Fid%3D638369916257545%26pin%3D4580">
      <tdStartTime>20:02:42.238</tdStartTime>
      <tdStartTimeMS>1399827762238</tdStartTimeMS>
      <tdElapsedTime>177</tdElapsedTime>
      <tdTotalElapsedTime>177</tdTotalElapsedTime>
      <tdStatus>200</tdStatus>
      <tdStatusText>OK</tdStatusText><tdRequestMethod>GET</tdRequestMethod>
      <tdContentSize>-1</tdContentSize>
      <tdMimeType>text/html</tdMimeType>
    <tdRequestHeaders>
      <tdRequestHeader name="Host">eturundus.eu</tdRequestHeader>
      <tdRequestHeader name="User-Agent">Mozilla/5.0%20%28Windows%20NT%206.1%3B%20WOW64%3B%20rv%3A29.0%29%20Gecko/20100101%20Firefox/29.0</tdRequestHeader>
      <tdRequestHeader name="Accept">application/json%2C%20text/javascript%2C%20*/*%3B%20q%3D0.01</tdRequestHeader>
      <tdRequestHeader name="Accept-Language">en-US%2Cen%3Bq%3D0.5</tdRequestHeader>
      <tdRequestHeader name="Accept-Encoding">gzip%2C%20deflate</tdRequestHeader>
    <tdRequestHeader name="X-Requested-With">
    XMLHttpRequest</tdRequestHeader>
    <tdRequestHeader name="Referer">
    https%3A//eturundus.eu/samsung-pin-lt/</tdRequestHeader>
    <tdRequestHeader name="Cookie">
    fbsr_265895733592477%3DMVuDCQySiQZErCLlQhy2Z-NISg3-pNSug5Hkw7Kxr58.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%3B%20PHPSESSID%3D58416520248001a68a259f52b7cf82dd</tdRequestHeader>
    <tdRequestHeader name="Connection">
    keep-alive</tdRequestHeader>
    </tdRequestHeaders><tdPostHeaders></tdPostHeaders>
    <tdPostElements>
    </tdPostElements>
    <tdResponseHeaders><tdResponseHeader name="Date">
    Sun%2C%2011%20May%202014%2017%3A02%3A17%20GMT
    </tdResponseHeader>
    <tdResponseHeader name="Server">
    Apache%20/%20DataZone
    </tdResponseHeader>
    <tdResponseHeader name="Expires">
    Thu%2C%2019%20Nov%201981%2008%3A52%3A00%20GMT
    </tdResponseHeader>
    <tdResponseHeader name="Cache-Control">
    no-store%2C%20no-cache%2C%20must-revalidate%2C%20post-check%3D0%2C%20pre-check%3D0
    </tdResponseHeader>
    <tdResponseHeader name="Pragma">
    no-cache
    </tdResponseHeader>
    <tdResponseHeader name="Keep-Alive">
    timeout%3D5%2C%20max%3D100
    </tdResponseHeader>
    <tdResponseHeader name="Connection">
    Keep-Alive
    </tdResponseHeader>
    <tdResponseHeader name="Transfer-Encoding">
    chunked
    </tdResponseHeader>
    <tdResponseHeader name="Content-Type">
    text/html
    </tdResponseHeader>
    </tdResponseHeaders>
</tdRequest>
</tdRequests>

Don't use a TStringStream at all. You are forcing UTF-8 decoding with it, but if the reply data is not UTF-8 encoded then reading the DataString can fail to decode the data and return a blank string.

Use the version of TIdHTTP.Post() that returns a String and let Indy decode the data for you based on the reply's actual charset:

function PostData(const AURL: string; AParamList: TStrings): string;
var
  _idHTTP: TIdHTTP;
begin
  _idHTTP := TIdHTTP.Create(nil);
  try
    _idHTTP.IOHandler := TIdSSLIOHandlerSocketOpenSSL.Create(_idHTTP);
    Result := _idHTTP.Post(AURL, AParamList);
  finally
    FreeAndNil(_idHTTP);
  end;
end;

Update: the screenshot you provided shows a GET request, not a POST request:

function GetData(const AURL: string): string;
var
  _idHTTP: TIdHTTP;
begin
  _idHTTP := TIdHTTP.Create(nil);
  try
    _idHTTP.IOHandler := TIdSSLIOHandlerSocketOpenSSL.Create(_idHTTP);
    Result := _idHTTP.Get(AURL);
  finally
    FreeAndNil(_idHTTP);
  end;
end;

procedure TForm1.BitBtn1Click(Sender: TObject);
var
  _Reply: string;
begin
  _Reply := GetData('https://eturundus.eu/samsung-pin-lt/_ajax/pin?id=276683492511635&pin=4580');
  ShowMessage(_Reply);
end;