I’m going to take a shot at this question since I’ve been pondering over it for some time, looking for the right [scientific] answer(s). Based on what I know, I think it is a combination of both psychological and biological reasons: cognitive neuroscience (perhaps some elements of behavioural psychology as well).

Apologies for my naive response, as a disclaimer I am not an expert so take what I say with a grain of salt.

It is biology...

Firstly, the “thinking” part of a long shower I believe is - as Hank stated - a result of a bombardment of stimuli from our sensory organs in our everyday lives. Obviously, our central nervous system (CNS) isn’t capable of responding to each and every one of the stimuli and consequently, we have developed (un)conscious filtering mechanisms - see selective perception. In the shower however, we aren’t subjected to this bombardment (or at least my showers aren’t) that has been exacerbated by the internet and social media in particular over the past couple of years. I suppose this is why we can be in a state of Zen and think better, inevitably losing track of time.

As the saying goes, a hot shower helps takes your mind off things (escape from reality?).

I don’t think that answers the question really well though.

If we unpack the first answer further, I suppose it is because showers are essentially hydrotherapy. It is a natural sedative due to the way our body responds to things like the temperature and the pressure exerted by the water (the hydrostatic effect of water generates a sort of “massage-like feeling”). These sensations are communicated through our nerves, to our CNS, and to the rest of our body - stimulating and invigorating various bodily activities, e.g. improving our blood circulation which may be a contributing factor to our improved ability to think.

Indeed, a study showed that “a moderately hot shower, could have a ‘crowding out’ effect on pathological processes within the mesolimbic system”. Simply put, [moderately hot] water does provide mental and physical relief by suppressing pain-related neurotransmission / negative stimuli. Cold water however, may depress activity of the frontal cortex (sedative effect). See pain modulations and mechanisms.

It is no wonder why we may often find showers so relaxing. It is therapeutic.

But, psychology!

I figured I should include a section about psychology as well because a. I am going to assume there will be anomalies and if so, psychology can account for the differences; and b. I like psychology.

Firstly, our behaviour and appreciation of standing “in the shower for a long time when thinking” may have been developed over time through psychological conditioning. The law of effect states that any behavior that has good consequences will tend to be repeated, and any behavior that has bad consequences will tend to be avoided. This is known as operant conditioning (learning based on the consequences of previous occurrences of the behaviour).

As I have explained earlier, the consequences of long showers may be the alleviation of stress (negative reinforcement) and/or the enhanced ability to think (positive reinforcement). If you find this desirable, the frequency of the behaviour being repeated will be increased. Think of it this way, you’re more likely to behave in class if it means getting some sort of reward from your teacher or reducing the likelihood of a punishment. In this case, you may resort to a [long] shower if it helps you escape all the stress caused by the stimuli you have received in your surroundings.

Before I conclude, I think it is worth mentioning the drive-reduction theory despite its pitfalls. The theory states that “humans are motivated to reduce the state of tension caused when certain biological needs are not satisfied”. Basically, we are motivated / driven to behave in certain ways by the active need to maintain a certain state of balance or equilibrium (homeostasis). Building on what I’ve already mentioned in the last paragraph, the reduction of the drive acts as a reinforcement for that behaviour. I may be overextending this theory for this question, but a long shower may satisfy our need to think or relax, and its success / reliability in doing so will reinforce our craving for it.

With that being said, will you stand in the shower for a long time when thinking if it was extremely cold or hot?

Suum cuique pulchrum est. To each his own is beautiful.

There may be an evolutionary explanation, but it is too fluffy and subject to a lot of criticisms.

To close, biology and psychology works hand in hand to explain the phenomenon that is showers and thinking.

Perhaps this can also explain why you think a lot when you are about to sleep or when you meditate / partake in yoga, Tai chi, etc.

I was tempted to say, science and psychology, but that would imply psychology is not a science which may be offensive. That is another debate for another day...

TL;DR: Read everything from the beginning.