Hi Alexander,
That's a really good question! There is no need to install the Dropbox desktop application on your computer, you can also access your files online on the Dropbox website if you wish. However, using the desktop application was some advantages as it is easier to handle files in your own Finder instead of online. If you'd like to safe space, you can use Selective Sync in order to decide which folders to sync to your computer:
https://www.dropbox.com/help/175
Please also note that on your mobile devices we don't save your files on the device's memory by default.
I hope this helps!