You're talking to your friend, who lives in another city. She asks about your baby, who she hasn't seen for several months. The baby talks now, so you say this.
She's talking up a storm these days.
"These days" means "recently". But "recently" is used to talk about things that have happened over the last few weeks or months. "These days" is a longer time period, usually from the last 6 months to the last 20-30 years. You use "these days" to talk about major changes in your life or in society.
I don't watch much TV these days.
Kids these days don't even remember what life was like before mobile phones and the Internet.
This is a casual expression that means to talk a lot.
You can use it to talk about young children who start talking a lot once they reach a certain age. Here are some other expressions that use "___ up a storm":
Mom would always be in the kitchen, cooking up a storm while we played in the yard.
You should have seen him at the wedding. He was dancing up a storm!
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