For IntelliJ (2022.1) onwards got to Settings -> Editor -> Inlay Hints -> Types -> Kotlin -> Types -> Parameter hints -> Local variable types or check all.
I think you are allowed to say, that this feature configuration is a little bit hidden in the menues and was moved a lot between different releases.

Thanks to @vitfo, adding this to the top here. For IntelliJ (2020.3) onwards got to:
Settings -> Editor -> Inlay Hints -> Kotlin -> Parameter hints -> Types
In IntelliJ 2019.3.1 onwards it is located at:
Settings -> Editor -> Inlay Hints -> Kotlin -> "Show parameter hints" (thanks to @zero).
Original Answer for IntelliJ 2018
Go to Settings -> Editor -> General -> Appearance -> "Show parameter name hints" -> Configure... -> Language: Kotlin -> Options -> "Show local variable type hints"