Because Kushina has a jutsu that can remove chains from her body. So that it can lock up Kyuubi.
When he was dying, Kushina told Minato that she wanted herself so that Kyuubi could be sealed into her baby who was none other than Naruto.
But Minato refused what Kushina said, he just wanted half Kyuubi in Kushina and other half in Naruto.
Minato then decided to use demon seal of death to seal half of kyuubi into Kushina body. And other half sealed kyuubi on Naruto.
Minato gave opening seal to one of Kuchiyose frogs and kept what little chakra he and Kushina had.
It is same as they can meet their children, when their children are older.
Both of them died after Kyuubi sealing.
Then Hiruzen Sarutobi who was third Hokage took care of Naruto even though villagers didn’t give it because of Kurama in Naruto body.
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