I'm just guessing, but I think it might simply just be for fun :)
Which one? They seem to do 600 calls / min, 5.000 calls / hour, 10.000 calls / day, 300.000 calls / month, how many times do you need to look up the weather for personal use? Fine, maybe you want 3 different locations, you can still call each of those sufficiently with those rate limits, no?
If you get an Arduino or Esp32 microcontroller (maybe in one of those starter-kits with various sensors), some breadboards, assorted jumper-cables and a kit with electronic components (resistors, caps) you'll be good to go. A device like a wall clock most likely won't require soldering, since it won't be jostled or moved around much.
Ben Eater: https://www.youtube.com/@BenEater/videos
Paul McWhorter: https://www.youtube.com/@paulmcwhorter/videos
Huy Vector: https://www.youtube.com/@huyvector/videos
I'd also take a look at the other DIY projects that people have linked in this discussion.
Seems like the author has experimented with 2 kindles side by side.
Microcontroller: FireBeetle 2 ESP32
Display: Generic 10" e-Paper display with driver board included
Timekeeping: DS3231 Real Time Clock Module
Temperature and humidity: BME280 module
Charging: Type-C USB 2S Li-ion BMS
That, along with a breadboard, two 18650 batteries, some resistors and capacitors make up the hardware. I modelled and 3D printed the case. I used the PlatformIO plugin (available for VSCode-based IDEs) for programming and transferring code to the esp32.
Weather API: https://openweathermap.org
For actual firmware I'd take a look at matada's github for inspiration (see the other reply in this thread). My own code isn't of the photogenic sort.