[Clare:] Henry is glancing around us, worrying that readers, co-workers are noticing us, searching his memory, realizing that some future self of his has met this radiantly happy girl standing in front of him. (1.1.5)
Clare realizes that the emotional connection she has with Henry in her present means nothing to him yet, because he met her in his future. She also finally gets that her strong emotions toward him, since so far she's just a "stranger," unsettle him.
[Henry:] And this astoundingly beautiful amber-haired tall slim girl turns around and looks at me as though I am her personal Jesus. My stomach lurches. Obviously she knows me, and I don't know her. Lord only knows what I have said, done, or promised to this luminous creature. (1.1.7)
Faced with Clare's beauty and strong emotions toward him, Henry fears that wherever and whenever he met her, he must have done something crucial to impact her in this way. He seems to feel guilty about it, which shows that he probably promised a lot of women a lot of things on his travels, because he thought he'd never see them again.
[Henry:] I realize that a massive winning lottery ticket chunk of my future has somehow found me here in the present, and I start to laugh. […] I see Clare running across Washington Square, jumping and whooping, and I am near tears and I don't know why. (1.1.7)
Although Henry has no specific memory of Clare, he knows that he will make memories with her in the future. This outlook triggers strong emotions in him he can't explain. It's almost as though his emotions remember how it feels being with Clare, even if the rest of his conscious mind doesn't.
[Henry:] It's funny how memory erodes. If all I had to work from where my childhood memories, my knowledge of my mother would be faded and soft, with a few sharp moments standing out. (1.6.1)
Henry reflects on how human memory fades and becomes inaccurate over time, preserving only the emotional highlights.
[Henry:] I stare at Clare, standing before me, and I am sorry to be here, sorry to ruin her Christmas. "I'm sorry, Clare. I didn't mean to put all this sadness on you. I just find Christmas… difficult." (1.6.40)
Whereas for most people Christmas is a time of joy to celebrate with family, Henry's idea of Christmas will forever be tainted by the memory of his mother's death and how it split his family apart.
[Clare:] "This is my room," I say apprehensively and Henry slips around me and stands in the middle of the rug just looking and when he turns to me I see that he doesn't recognize anything; nothing in the room means a thing to him, and the knife of realization sinks in deeper. (1.9.44)
Clare realizes that in Henry's present, the whole world she's built with him doesn't yet exist – he was an older man when she met him before. She sees that the strong emotional bond she has with him is – as of yet – a one-way street.
[Clare:] Sometimes I would give anything to open up Henry's brain and look at his memory like a movie. I remember when I first learned to use a computer; […] I wanted to push my hands though the screen and get the real thing in there, whatever it was. I like to do things directly, touch the textures, see the colors. (1.9.44)
Clare muses on how memories are intangible, which makes her wish she could somehow touch them or grasp them – the act of touching things makes them become real to her.
[Henry:] Today is the thirty-seventh anniversary of my mother's death. […] If fervent memory could raise the dead, she would be our Eurydice, she would rise like Lady Lazarus from her stubborn death to solace us. (2.28.3)
Henry's memories of his mother are so strong and immediate that it's difficult for him to believe that she's dead.
[Clare:] I am conducting conversations with Henry as though he were here with me, as though he could see through my eyes, think with my brain. (3.3.8)
Clare misses Henry so much that she conducts conversations with him about the present so she can keep sharing her life with him and continue to create memories with him.
[Clare:] It is only memory that holds me here. Time, let me vanish. Then what we separate by our very presence can come together. (3.4.5)
It's Clare's memory of her past with Henry that keeps her waiting for him, grieving over him. With no consciousness of the past, she'd be able to move on.