I see a lot of complicated answers, while this is super simple in Bootstrap 3:
Step 1: Use the official example code to create your radio button group, and give the container an id:
<div id="myButtons" class="btn-group" data-toggle="buttons">
<label class="btn btn-primary active">
<input type="radio" name="options" id="option1" autocomplete="off" checked> Radio 1 (preselected)
</label>
<label class="btn btn-primary">
<input type="radio" name="options" id="option2" autocomplete="off"> Radio 2
</label>
<label class="btn btn-primary">
<input type="radio" name="options" id="option3" autocomplete="off"> Radio 3
</label>
</div>
Step 2: Use this jQuery handler:
$("#myButtons :input").change(function() {
console.log(this); // points to the clicked input button
});