Yoshimoto's novel Kitchen is about Mikage, who is bereft and alone when her grandmother passes away. She is invited by Yuichi Tanabe to go and live with him and his mother, Eriko; she agrees and moves in with them, finding comfort in the kitchen. The kitchen is not only a symbol of comfort for Mikage, it is a place where she can be honest with herself about her own challenges and struggles in life. You asked about some examples in the text which reference what the kitchen means to Mikage:
1) "I love even incredibly dirty kitchens to distraction-vegetable droppings all over the floor, so dirty your slippers turn black on the bottom."
The reason Mikage is not bothered with dirty kitchens is plain: it is a distraction from her grief and the pain of being alone. The kitchen offers her comfort and a sense of normalcy. "Now, only the kitchen and I are left. It's just a little nicer than being all alone."
2) "...I want to breathe my last in a kitchen. Whether it's cold and I'm all alone, or somebody's there and it's warm, I'll stare death fearlessly in the eye."
The kitchen not only comforts her, it also gives her courage to face her fears. She is not afraid to face the reality of death if she can do so in a kitchen, a place that has helped her forge her own identity and given her a reason to continue living.
3) "It was a good kitchen. I fell in love with it at first sight."
Mikage says that she feels "an immense loneliness" when she first comes face to face with people she barely knows. Here, the Tanabe's kitchen has welcomed her with open arms and she is comforted that she will find her place in the strange home she has been invited to live in: "It was clear that in spite of the disorder, everything was of the finest quality." She sleeps next to the kitchen, but this time she isn't lonely. She says that she is "too sad to be able to sleep in the same bed with anyone; that would only make the sadness worse."
4) "It was a kitchen. I was puzzled, smiling about how I had just gone from the darkest despair to feeling wonderful."
When Mikage gets off the bus, she is overwhelmed by her emotions and cries for the grandmother she will never see again. When she looks up, she sees a kitchen overhead where she hears happy voices and pots and pans clanging. Once more, she finds her courage.
5) "Why do I love everything that has to do with kitchens so much? ...Perhaps because to me, a kitchen represents some distant longing engraved on my soul."
Mikage talks about learning how to cook and how she becomes an assistant to a cooking teacher. She learns that her one great flaw is "lack of precision." In her obsessive persistence to perfect the art of cooking, she comes to make peace with the one thing she knows about life: that all of us are really alone and must face the reality of our approaching deaths alone. Mikage finds her salvation in her cooking, in her kitchens; this is what gives her courage to "continue living with the awareness" that she will die.
Hope these examples are helpful. You will likely find more within the novella, but these quotes might provide a starting point. Thanks for the question!