It’s very strange that even though there are millions of developers who build applications for Windows computers they are slow to move those applications to Windows phones.
Fortunately, Dropbox did finally release an application native for Windows 8 phones that you can use to interact with Dropbox. It came out early in 2015 and should be available for you to download.
If you don’t want to use that application, you are always welcome to use a web browser and access DropBox through that web browser. But as of the beginning of the year you can access DropBox using a native Windows 8 phone application from Dropbox directly.
DropBox for Windows 8.1 Phones.