The ATtiny10 programming uses a protocol called TPI to write the firmware into the flash memory. The cheapest way to program an ATtiny10 is either by Arduino or USBASP. Keep in mind the USBASP needs to have the latest firmware. A tutorial on how to update it is here. A detailed guide to program the ATtiny is here.

You can either get the code and compile it or download the firmware. Both are on GitHub. To upload the code you can use avrdude:

./avrdude -e -v -p attiny10 -C ./avrdude.conf -c usbasp -U flash:w:firmware.hex:i -P usb 

 Pay attention to the path of the avrdude configuration file.

The connections are very simple. Just follow the schematics.