To the OP, the instructor does not kill the engine, but jut pull it to idle to simulate engine failure.
Not to hijack this thread either, but why is spin recovery not trained anymore? When I did my pilot training in the 90's, in Canada, (Private, Commercial, Instrument, and Multi Engine) we did spin training. I almost got sick the first few times but after I soloed and flew by myself for some time I found myself doing spins all the time (at 10,000 feet of course) and liked it. It was actually fun. Wasn't allowed to to do it with passengers though

I know the US does not do spin training for private pilots (I think) and I think this should be required training to get your pilots license.
I can't tell you much about US regulation, but in the U.K. spin training hasn't been mandatory for some years.
The reasons for this are threefold! Firstly, many light aircraft, including some trainers, are not approved for deliberate spins. Piper Warriors spring to mind, which will probably be familiar to U.S. readers, but there are many others!
Secondly, many more people were being killed training for spins than in real spin accidents. A former colleague nearly killed himself spinning a few years back (just because it isn't compulsory doesn't mean it isn't taught). It transpired the C of G was an inch out of limits, but that was enough to delay recovery by nearly 4000 feet!
Finally, it's almost impossible to spin (properly) and recover in much under 800 feet. Most real spin accidents occur on the turn to final approach when an aircraft is seldom above 600 feet, and any spin at this point is usually fatal. I have personally witnessed two such incidents, both with that outcome.
The regulators have therefore decided, quite correctly in my opinion, that the emphasis should be on 'spin avoidance' rather than 'spin recovery'. The considerable reduction in spin related fatalities since this policy was introduced more than justifies it in my opinion!
Of course, if you want to practice spins, or aerobatics, or whatever, then any instructor will be delighted to allow you to do so, but in a suitable aircraft, in suitable weather conditions, and probably at a stage in your training when you have a suitable amount of experience to benefit from the manoeuvre. In my experience this is usually at 30 hours plus, as part of your test preparation, and not at 10 hours prior to first solo, which is when it was historically taught!