Well the lens itself will let in the same amount of light on either camera. The thing is that the sensor in the D600 is much larger than the one in your D3200, roughly about two and a half times larger. As a result it's more sensitive to that light and able to do a lot more with it. To put it in terms of Fstops, your looking at roughly a 1.4 Fstop difference.
Can you explain it a bit better ? If there is the same intensity of light per every square millimetre of the sensor how comes bigger sensor should sense more light ?
Well I can.. problem is doing so in such a fashion that the post doesn't become longer than Tolstoy's War and Peace.. lol. Ok, I'll give it a quick stab, and no this is by no means complete but hopefully it will cover the basics.
A sensor in a digital camera has millions of spots that are sensetive to light, called photosites. Each photosite records information about what is being "seen" through the lens, so the more photosites you have and the larger they are naturally the more information that is being collected. A larger sensor allows for larger photosites - and a larger photosite will collect more from a ray of light than a smaller one can. If it helps think of it in terms of water. If you put a coffee cup on the floor and pour water into it, once the cup is full any more water will flow out of the top and simply be wasted. Replace the cup with a bucket, and now you can pour the same amount of water and because of the buckets larger size, it will collect a lot more of the water.
The amount of water your pouring doesn't change (much like the amount of light coming into the camera doesn't change when your using the same aperture) however because you now have row after row of buckets instead of row after row of coffee cups, the amount of "water" or in this case the amount of information from the light captured that is retained is a whole lot more.