If you want to find when you started a walk which you have completed, that's easy - just login to your account on the website, go to My Walks and then click on Completed Walks.
To find out when you started your current walk is not so straightforward. To find this out, click on the View More button at the bottom on your Activity feed on your dashboard and then search back through the pages of your activity. There will be an entry showing when you started your walk. The entry will only refer to how many days, weeks or months ago that was, depending on how long you've been doing the walk.