Kitty has been ill all winter, and her health has been growing steadily worse as spring approaches. The family doctor has already tried all of his traditional methods, and nothing has helped. He recommends that she visit a famous doctor from abroad.
Here, Tolstoy gets in a little jab at the medical profession: he notes that the doctor, a not-yet-old man, insists that maidenly modesty is barbaric and insists on examining Kitty naked. He does this every day as part of his practice, and can see nothing wrong with it because he's not uncomfortable.
The doctor prescribes a trip abroad to take the waters at a famous spa—not because he thinks it'll necessarily help, but because it'll do no harm, and Kitty's mother wants to go.
Kitty's father believes he's the only one who knows why she's ill, and he doesn't believe anything the doctor says. Kitty's mother has the opposite point of view, taking the doctor's word as gospel because it's science.
Kitty's mother cheers up at the doctor's words, and Kitty pretends to be cheered, too. Underneath is all, though, she's still miserable from heartbreak.