Six didn't want Claire to misunderstand him or be afraid. He needed to find the right words to reassure her but wasn't positive he was capable with his limited emotional vocabulary. What could he say to prove he wasn't a threat to her?
His reasons for protecting her weren't as flimsy as she thought. Originally it only had been out of curiosity but now it was because he genuinely didn't want her to die. As the first person to ever pique his interest, he wanted her to stay alive.
Now that going back to headquarters wasn't an option—even if he hadn't killed five of their agents, the outside was far too interesting to leave—he would need a guide here. A guide who knew about his past so he could ask things freely. Claire was the only person who fit that description.
On top of that, Six liked her. He didn't want something he liked to be taken away from him. His first and currently only friend was worth protecting even if it was inconvenient.