First go to the Download manager on blesta.store/portal, click Downloads and then click BlestaCMS > Stable and download the latest version.

Upload the zip to the root of Blesta for example we'll be using the BlestaCMS Pro for our website so Blesta is installed in: /home/username/public_html/

Extract the zip and the files should be in their correct location. If you wish to double check see that these are correctly installed:
/app/views/client/cmspro3/
/app/views/client/cmspro4/
/plugins/cmspro/

Now head over to settings > company > plugins and click on Install next to the BlestaCMSPro.