Before the challenging times, Jesus prepped the disciples on said challenging times and elaborated on various promises. Jesus also talked about how although the disciples would be physically separated, eventually "[the disciples] will see [Jesus] because [Jesus] go[es] to the Father" (16:16). In other words, the separation that the disciples would face would only be temporary.
However, some of the disciples did not understand what Jesus was referring to -- especially when Jesus said "a little while". They wanted to ask Jesus what he meant by "a little while".
In response to the disciples' question, Jesus told the disciples that while "the world will rejoice", the disciples "will weep and lament", but "[their] sorrow will be turned into joy" (16:20). Jesus prophesized the disciples' emotions likely from current emotion projections, and he knew that the disciples would have a hard time dealing with being separated from Jesus physically -- especially with the world throwing pressure to them. However, because Jesus "will see [them] again", Jesus promises not only that "[their] heart will rejoice" but also that "[their] joy no one will take from [them]" (16:22). In other words, eventually, the disciples will get joy that lasts, and the disciples experienced said joy after Jesus's resurrection (especially in heaven).
For comparison, Jesus used an analogy of a pregnant woman. In many cases, a woman has to undergo labor pains during pregnancy (especially in the ancient days). However, once the child is born, the woman generally experiences joy upon seeing the child, even though the woman had to go through labor pains. In a similar way, while the disciples would experience great sorrow (especially from physical separation from Jesus), the disciples will eventually get lasting joy (especially since Jesus would take on the sin penalty on behalf of humanity while He is away).