Mostly known from the man himself - Wim Hof - cold showers are a great way for your immune system.
While he goes obviously deeper into the actual physical response and why it matters for your body, there are also some other aspects why I do it.
How do I actually shower cold?
Tbh, I always shower normal warm, not hot as in like the mirror is gonna be misted up. When I am at home that is mostly it. However, I am at the gym ~5 a week and when I am there, the shower is nicer and there I always end with a cold shower.
What do I do?
After finishing the actual cleaning, I will turn the water a bit colder, fell the water, bit colder, feel it and so forth until it is actually cold, like muscle tensing and not following the instinct of just jumping out of the shower.
Why I do it?
1. Getting out of your comfort zone, showing yourself you can do more
Every time I shower cold, it is making me feel uncomfortable. But also, every time I shower cold, I show myself that I can push outside my comfort zone. That I am in control of what my body does or not. Doesn't matter how much it tells me that it's cold. I am not going to die from these 30-120sec, so I will remain under the water.
2. Meditation
What helps me actual go through it, is sort of disconnecting from my body. I take the physical response and just observe it. "Aha, now my muscles are tensing, breathing becomes harder and shorter, ..." - taking the role of an observer makes me actually curious and is a meditative state that you can reuse in other life situations. You are not your body.
3. Humbling
No matter how much money you have in your pocket (I don't have much, but even if I would), it's a leveling experience. Not matter who, money, fame etc doesn't matter. It's only you and your body/mind against the cold, nothing else matters.