It's interesting they only focus on the cold shower. I would have thought it would be short and barely give you enough time to get over the initial shock(cold shock response) - around a minute or so when the gag reflex hits and what would potentially drown you in open water. After that shock, things change.
My personal experience is with swimming in the sea year round for 3 to 4 years. 2 to 3 times a week every week. I started in August to help acclimatise to the change in the temperature as it cooled. Temperature ranges from 18C in August to 4C in early March, give or take.
The effects were great over all and I thought I was healthier for it. I understand the tightening of the muscle which some of the others mentioned as an experience. I got used to this and it seemed to relax me. I never really got the tight muscles thing except afterwards when I kind of wanted to tighten up.
I would walk in at a reasonable pace. As others have said, with a meditative feeling. Once fully emerged and the shock had passed I could swim for a maximum of around 5 to 10 minutes at the coldest part of the year. Sometimes it was more a splash around for as long as I could stand it. I generally stopped and got out when my wrists started to cramp. I'm guessing the cold water would send the blood circulation internally away from the skin.
It always felt the warmest in late September early October for some reason. The coldest place I ever swam was between two San Juan islands in August!
Once out of the water I would feel warm and have a feeling of glowing. Drying off would be a pleasure despite the outside temperature. My overall demeanour would be very positive and up lifting. I've heard talk of endorphins or something along those lines being released in to the body.
I never jumped in to a hot shower right after preferring to warm naturally and slowly. I felt warm for a good twenty minutes after getting out and being dressed fully whoever cold it was.
The whole experience became slightly addictive which is why I did it so long. Work has now gotten in the way of that.
There is a serious safety aspect to all this and it should be done with caution and understanding. This guy, Mario Vittone, was very informative on what's going on when trying to stay alive in cold water: http://mariovittone.com/2010/10/1-10-1/