Why can I not add or remove any files?

I can't do anything except view / review

It sounds like you were added to another person's project or account as a Viewer or Reviewer. If so, you will only be able to view/review within that Project, whether or not you pay for your own account.

Regardless of the plan you're signed up for, you can still be invited as Admin/Member to someone else's paid account. If you need to participate actively in the Project by uploading or managing projects and media, you will need to be added to this Project as Member or Admin respectively.

Currently only paid accounts allow for more than one team member.  This means that whoever manages the project you need to work on needs to have a paid plan.

And if they don't yet, we offer a 14 day trial ;)

I'm definitely an administrator in this workspace, but I still can't do anything.

If you opened a review link from someone on your team while not already logged in, and then selected Join as Guest instead of logging into your account in the dialog shown below, you will not be able to do anything other than participate in the review.  

If this is the case, go to the default user icon in the upper right corner and log outSign In with your SyncSketch administrator credentials, then open the review link again.

To learn more about Workspaces, projects, inviting users and their permissions, see Roles, Permissions & User Access Management