If your intent is identification, the Identify page is the best place, since all of your observations are excluded (as long as you’ve added an ID to it) and it will not show you, by default, the observations by others you’ve already added IDs to: https://www.inaturalist.org/observations/identify?place_id=12&taxon_id=48139
But if you definitely want to use the Explore page, you can use
not_user_id= like @alexis_orion says, but instead of the username, for some reason you have to use the user number, so: this excludes yours: https://www.inaturalist.org/observations?place_id=12&subview=grid&taxon_id=48139¬_user_id=593075 (you are @salticidude?* Awesome photos, by the way!) vs. all Salticidae in Oklahoma at https://www.inaturalist.org/observations?place_id=12&subview=grid&taxon_id=48139
You can find your user number in the main header, top right corner, hover over your icon and click Profile, then check the URL. Not sure a more convenient way to find someone else’s user number, but I usually go to their profile and hover over the Follow/Stop Following button and check which URL that would take me to – their user number is displayed when you hover there, at least in the lower left corner in Chrome/Firefox.
*It’s best if your iNat Forum account matches your username on iNat; you can edit your username on the forum here: https://forum.inaturalist.org/u/bothrops07/preferences