While the above answer is correct, for the curious here's a more technical answer on how to control the secondary Climber:
Assuming that Popo is the lead and you want to control Nana. You control Popo as you would a normal character, and when close to him, Nana performs moves with a roughly 1/10th of a second delay, as if the input was issued a bit late to her. Generally speaking, whenever Nana is not within close range, her AI kicks in and does actions in a fairly predictable manner.
When you are preoccupied by an action and press a button, the input is buffered. If that action ends within ten frames (1/6th of a second), the buffered input is executed and you will perform the action specified by it without delay. Due to the timing difference between the two Climbers, it is possible to buffer a move in a way that only Popo will execute it, allowing you to de-sync the Climbers' movements and do completely different actions with them. This can be utilized to essentially control them separately as long as they are close to each other.
It can sometimes happen by accident, and you might for example see only Nana perform a special move, even though normally both Climbers would do it. However, it is fairly uncommon and any prolonged de-sync between their actions require deliberate and accurate movement.
However, there are some situations where you are able to control Nana separately from Popo. Most notably, when Popo is grabbed or holding the opponent, and Nana is within range, it is possible to issue commands to her without Popo being able to move. This could lead to an illusion that you are controlling Nana directly. Similarly, grabbing a ledge renders Popo unable to move for a moment, while Nana can still do actions if she is close enough and not grabbing the ledge herself.
Additionally, there is a very short period of time after Popo losing a stock where Nana can be controlled at will, regardless of distance. You can see an example of the behavior in this video, where the Ice Climbers player was able to use forward air as Nana on command after Popo had died. Note that Nana's AI will never try to use forward air while off-stage and close to the ledge.
There are also some very specific situations where Nana can be controlled directly, but they are difficult to pull off even deliberately, are extremely rare in top-level competetive play and most likely never come up in casual games. The situations described here do happen often though, and a good understanding of their application is essential for competetive Ice Climbers players.