If you used the previous version of WeTransfer, you'll know that downloading worked pretty much the same as uploading. For technical and reliability reasons, in the new version downloading now works differently. We now download all files through the browser, so you will no longer see the progress in WeTransfer, and you may not be asked where to store the file. The download progress should be visible at the bottom of your browser window (in Chrome and Internet Explorer), in the Downloads window (Firefox or older Safari versions), or in a little button in the top right corner of the window (newer Safari versions).
In most cases your file will stored in your computer's Downloads folder. This can be changed (instructions for Chrome, Firefox, and Safari), and Chrome and Firefox also gives you the option of asking every time where to save the file.