So I went to Stan after the whole events of The Author about that picture I found in the basement.
Stan explained it. He said, after I brought Mabel to hear it too, that when Mabel and I were born, Grandpa Shermie invited Stan to come see us. Stan said, and I quote, "I took my car and drove as fast as possible, got a few speeding tickets, but who cares! I wanted to see my grand niece and nephew! I ran as fast as possible to the hospital room your mother was in. Sure enough, your grandpa Shermie was there. I miss, Little Sherm. Anyway, your mom let me hold you two. Ever since Mabel's first 'myah,' I didn't want to let you two go.
"In fact, I actually tried to take you home with me, believe it or not. All the doctors, Sherm, and you pa chased me all the way outside the hospital. If I didn't trip on my shoelace, then I would've gotten away with you too."
Mabel thought it was sweet that he wanted to keep us. I was more concerned if he dropped us. He said that he actually didn't and that he actually "twisted my body in a way that permanently ruined his back so we wouldn't hit the floor. That's when Dipper went, 'bwah.' That's REALLY when I wanted to keep you but Sherm wouldn't let me hold you anymore, because I was 'too clingy.'"
Grunkle Stan is such a sweet guy! I thought he hated us, honestly. He treats me harsher than Mabel. Sure, I already know why, he wants me to be stronger, 'cause he seems myself in him as he explained in his mind, but it's still super sweet. I guess that also explains why his back hurts so much.
Author's Note
This is one of my favorite chapters. Canonly, Stan does get to see baby Dipper and Mabel (according to his dialogue in The Legend of the Gnome Gemulets (I've never played but I saw pictures)) but I don't know if we actually get the story and I've always wanted a story. I think it wouldn't been cute if Stan was a sweet guy with his baby niece and nephew so that's where this came from.