We download all files through the browser, so you will not see the download progress in WeTransfer itself, 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 store in your computer's Downloads folder. This can be changed (instructions for Chrome, Firefox, and Safari). Chrome and Firefox also gives you the option of asking every time where to save the file.
If you are struggling to download the transfer, you can take a look at this article: "How do I download a transfer that was sent to me?".