MacOS doesn't fit external screen and I don't know how to stretch it Model: MacBook Pro (Retina, 13-inch, Late 2012)
Graphics: Intel HD Graphics 4000 1536 MB
Monitor: Samsung U28E590D 28-Inch UHD
Using HDMI cable to connect to the monitor.
Things I have tried:

Option + click to see all the possible resolutions
Reset NVRAM
Reset SMC
Using SwitchResX
Playing the the monitor's settings

Nothing works! I have absolutely no idea how to fit the OS to the screen. The text also looks a little blurry.
How it looks like: https://imgur.com/2ZZx68D
Help on fixing this would be very appreciated!
Short answer, you can't
If you have a look at the specs on the monitor It's a 4K monitor 3840 x 2160.
Then have a peek at the specs on your computer. I usually use EveryMac.com.
And under Second display support on the second link above it says:

*This model supports a simultaneous maximum resolution up to 2560x1600 on two external displays via Thunderbolt. Alternately, it can support
a single display up to 2560x1600 via Thunderbolt and a single display
up to 1920x1200 via HDMI. Although it can theoretically power all
three external displays, as confirmed by a helpful reader, it runs too
hot with three displays connected.

To me that says your Mac does not support the highest resolution of that monitor, so you are going to get a resolution that is lower and of poorer quality.
You need to get a monitor that supports 2560 x 1600 native resolution for optimal display output.