You can't do it on the AU site.

The way I remove them is to open another browsing window and login to ebay.com. On the home page you'll see Recent searches and Recently viewed on the left hand side. Click on Clear next to both of them. To check that they're gone, switch back to your ebay.com.au window and click on the ebay logo to refresh.

Just do some other searches for things that you don't mind other people seeing and they will take their place. I often do this when I'm searching for presents for others in the house.