Why do I get attacked by Thalmor Justicars in Skyrim when I have done nothing to offend them as far as I know?

It seems there are a number of reasons you can be spontaneously attacked by Thalmor Justicars. As the comments have mentioned, it may be related to your actions during the quest In My Time of Need. The wiki for that quest states:

If Kematu's request is accepted, and the Dragonborn helps him capture Saadia, the Dragonborn may encounter a group of Thalmor Justiciars in the wild that carry a Justiciar Execution Order with the Dragonborn's race and name.

Afterwards, it clarifies when these orders are commonly given:

Execution Orders are normally issued because of the assault on the Thalmor Embassy in the main questline during the quest Diplomatic Immunity. Thalmor Justiciars that carry a Justiciar Execution Order also appear after attacking and killing any wandering Thalmor agents.

The wiki on Thalmor Justicars states:

The Justiciars may also attack immediately if the Dragonborn is wearing an Amulet of Talos.

I suspect that last reason would not include the note you mentioned, but it is another reason why you might be attacked in a seemingly unprovoked manner.

After completing 'Darkness Returns' main quest for the Thieves guild will also cause the justicar to come after you. I noticed an execution note on one right after I left the Twilight Seplicar. The note stating that I am an enemy and to destroy me with extreme caution and that I'm capable of fending for myself so be carful. So I assume I'm wanted now by them ha ha

It's a random event that happens after Level 8. It IS a bit odd that it happens for no reason, but it is just a random event.

If you help the Gray-Manes in the quest Missing In Action, the Justicars will come for you at some point soon in the future. Even if you never assault Northwatch Keep, simply helping gather the evidence will brand you a wanted man with the Thalmar.