Why do smartphones slow down? What happens inside?

Operating System updates.
Simply put, updates to the OS are made to run on the newer hardware from the manufacturer, by updating you're older device, you're increasing the workload on it's hardware, some features which are not even usable on the current device.

Application updates.
Developers continuously update their apps adding features and security. Some of these might not even be necessary in your day to day life, leading to slow downs in terms of performance.

Memory Space.
Phones use NAND technology that has a certain number or read write cycles that once passed the manufacturer cannot predict the rate of degradation. Also, it utilises a specific amount of free memory blocks in order to work at peak performance and once you clog up the space, it minimizes the performance.

Background applications.
These are the devil's in today's computing. Certain processes and apps run in the background without knowledge and as more and more apps are added, this mutliplies their numbers. These can take up a lot of the RAM and cause certain apps to slow down as they don't have enough leeway to operate at peak.

Battery Life.
Like the Apple case whereby the processor peak voltage is reduced in order to prioritize battery life, this is because the battery is the weak link in electronics, it is the fastest to decay.

Perception.
When a person looks at a new phone and sees how fast it is is, going back and looking at you're older phone puts the idea in you're mind that the phone has slowed down.