YouTube – Improve Auto-Generated Subtitles/CC for Videos

Use "Network inspector" function of your browser's script debugger and find second request to timedtext page after enabling transcribed subtitles then just copy all that request to the addressbar to download them in native YouTube xml format.

To get SRT version run this code in the debugger console for that xml's page:

function makeTimeline (time) {
    var string, time_array = [], milliseconds = Math.round(time % 1 * 1000).toString();

    while (3 > milliseconds.length) {
        milliseconds = '0' + milliseconds;
    }

    time_array.push(Math.floor(time / (60 * 60)));
    time_array.push(Math.floor((time - (time_array[0] * 60 * 60)) / 60));
    time_array.push(Math.floor(time - ((time_array[1] * 60) + (time_array[0] * 60 * 60))));

    for (var i = 0, il = time_array.length; i < il; i++) {
        string = '' + time_array[i];
        if (1 === string.length) {
            time_array[i] = '0' + string;
        }
    }
    return time_array.join(':') + ',' + milliseconds;
};

function returnSRT (data) {
    var caption, previous_start, start, end, temp, captions = data.getElementsByTagName('text'), srt_output = '';

    for (var i = 0, il = captions.length; i < il; i++) {
        caption = captions[i];
        start = +caption.getAttribute('start');

        if (0 <= previous_start) {
            temp = captions[i - 1].textContent.replace(/</g, '&lt;').replace(/>/g, '&gt;');
            srt_output += i + '\n' + makeTimeline(previous_start) + ' --> ' + makeTimeline(start) + '\n' + temp + '\n\n';
            previous_start = -1;
        }

        if ( end = +caption.getAttribute('dur'))
            end = start + end;
        else {
            if (captions[i + 1]) {
                previous_start = start;
                continue;
            }
        }

        temp = caption.textContent.replace(/</g, '&lt;').replace(/>/g, '&gt;');
        srt_output += i + '\n' + makeTimeline(start) + ' --> ' + makeTimeline(end) + '\n' + temp + '\n\n';
    };
    return srt_output;
}

returnSRT(document.documentElement)

Also here is bookmarklet version of the script:

javascript:(function(){function%20makeTimeline(time)%7Bvar%20string%2Ctime_array%3D%5B%5D%2Cmilliseconds%3DMath.round(time%251*1000).toString()%3Bwhile(3%3Emilliseconds.length)%7Bmilliseconds%3D'0'%2Bmilliseconds%3B%7Dtime_array.push(Math.floor(time%2F(60*60)))%3Btime_array.push(Math.floor((time-(time_array%5B0%5D*60*60))%2F60))%3Btime_array.push(Math.floor(time-((time_array%5B1%5D*60)%2B(time_array%5B0%5D*60*60))))%3Bfor(var%20i%3D0%2Cil%3Dtime_array.length%3Bi%3Cil%3Bi%2B%2B)%7Bstring%3D''%2Btime_array%5Bi%5D%3Bif(1%3D%3D%3Dstring.length)%7Btime_array%5Bi%5D%3D'0'%2Bstring%3B%7D%7Dreturn%20time_array.join('%3A')%2B'%2C'%2Bmilliseconds%3B%7D%3Bfunction%20returnSRT(data)%7Bvar%20caption%2Cprevious_start%2Cstart%2Cend%2Ctemp%2Ccaptions%3Ddata.getElementsByTagName('text')%2Csrt_output%3D''%3Bfor(var%20i%3D0%2Cil%3Dcaptions.length%3Bi%3Cil%3Bi%2B%2B)%7Bcaption%3Dcaptions%5Bi%5D%3Bstart%3D%2Bcaption.getAttribute('start')%3Bif(0%3C%3Dprevious_start)%7Btemp%3Dcaptions%5Bi-1%5D.textContent.replace(%2F%3C%2Fg%2C'%26lt%3B').replace(%2F%3E%2Fg%2C'%26gt%3B')%3Bsrt_output%2B%3Di%2B'%5Cn'%2BmakeTimeline(previous_start)%2B'%20--%3E%20'%2BmakeTimeline(start)%2B'%5Cn'%2Btemp%2B'%5Cn%5Cn'%3Bprevious_start%3D-1%3B%7Dif(end%3D%2Bcaption.getAttribute('dur'))end%3Dstart%2Bend%3Belse%7Bif(captions%5Bi%2B1%5D)%7Bprevious_start%3Dstart%3Bcontinue%3B%7D%7Dtemp%3Dcaption.textContent.replace(%2F%3C%2Fg%2C'%26lt%3B').replace(%2F%3E%2Fg%2C'%26gt%3B')%3Bsrt_output%2B%3Di%2B'%5Cn'%2BmakeTimeline(start)%2B'%20--%3E%20'%2BmakeTimeline(end)%2B'%5Cn'%2Btemp%2B'%5Cn%5Cn'%3B%7D%3Breturn%20srt_output%3B%7Dwindow.location.href%3D'data%3Atext%2Fplain%3Bbase64%2C'%2Bbtoa(returnSRT(document.documentElement))})();

Only some channels allow users to contribute subtitles.

From Contribute subtitles and closed captions - Youtube Help

Some channels let you contribute subtitles and closed captions to their videos. You can submit content for an entire video or just add what you know — every little bit helps make videos more accessible to a larger audience.

If the video's creator accepts your submission, they'll be viewable on the video by clicking on the [CC] icon in the player. Approved content is owned by the video owner, but your YouTube username and public profile may be visible to the video uploader and other video viewers.

Before you start: Make sure the video you want to add captions or subtitles to is part of a participating channel.