Hey @andrea941, welcome to our Community.
As long as you've been added as a member to this Dropbox Business team, you don't need to purchase your own space to access the files.
Do you know where the files are currently located in the admin's account?
Is it possible that they're just in the folder with their name on it? That would be the 'team member folder', so only the person accessing that account would be able to access the files.
If the files are in a team folder in the team space instead, you should be able to access them, if the admin has given you access.
You can read more on how to go about sharing files in a team environment here.
Let me know how it goes!