Simple answer is Azure Blob Storage will not do the unzipping for you. This is something you would need to do on your own. How you do it is up to you.
One possibility is (like you mentioned) that you upload zip files on your server, unzip them there and then upload individual files.
Another possibility is to do this unzipping through a background process if you are concerned about the processing happening on the web server. In this approach, you will simply upload the zip files in blob storage. Then through some background process (could be WebJobs, Functions, Worker Roles, or Virtual Machines), you would download these zip files, unzip them and then re-upload these individual files.
To trigger the background process on demand, once the zip file is uploaded you could simply write a message in a queue telling background process to download the zip file and start unzipping process.
