Why does it jump one character to the right after running the script?

The object app.selection[0].contents has a string length of seven characters, set by the initial selection. Your code increases the contents without increasing the length (7 characters, in this case).

Try this:

if (app.selection.length > 0 && app.selection[0].hasOwnProperty('contents')) {
    var selectedText = app.selection[0];
    var originalText = selectedText.contents;
    selectedText.contents = '[' + originalText + ']';

    // Adjust the selection to include the new brackets
    var startIndex = selectedText.insertionPoints[0].index;
    var endIndex = selectedText.insertionPoints[selectedText.insertionPoints.length - 1].index;
    selectedText.parentStory.characters.itemByRange(startIndex, endIndex + 1).select();
}