You don’t put Hugo on the server (well, you could but that’s a whole other topic), you upload the content of the ‘public’ folder that Hugo generated when you tell it to build your site.
You can upload those files via SFTP or whatever tools your host provides to the relevant directory in your hosting account - just like where you’d stick Wordpress’ files.
Those files from the public folder ‘are’ your site, it’s a static site generator after all so it outputs all the HTML/CSS/JS/media files that you need to that public folder and you simply upload them.
When you update the site, you re-upload the files (I use rsync to do it so it does it intelligently and doesn’t re-upload stuff that hasn’t changed).
That’s a pretty simple overview but should hopefully help.