Subplots are a thing that gives a game a rich and vibrant feel. They assist in making the world more real to a character. They are also a super handy tool for the GM to involve the players and think you are a genius. So how does the subplot work?

In the sandbox, all your plots are subplots
If you have built a sandbox style game then all of your plots are laid down as part of that. The players will likely choose one of these plots and turn it into the main plot. But they would have heard about other things going on in the game. The spacer in the bar talking about the excise tax because of pirates. The bartender who was poisoned with the phage from a woman who called herself the Nightgaunt and wants to find her. The group of gentlemen seeking the shape-changing Doppleganger who killed their companion who goes by the name of Danien. All of these things are there for the players to register. They may choose one to become their major focus but that allows you to add to these others along the way.
Of course, if your world is not designed in a sandbox way you can still have a book filled with plot details like this that the players can hear about. It just requires a little bit of extra work as you play to thread the details in.
Subplots are things of beauty
Subplots are a games master’s (GM) friend. If you are good at taking notes you can add to the subplot at your leisure. If it is a sandbox then you probably advance all of these plots as you plan. You will know how and when the players may pick up on developments. In a non-sandbox style game, you can choose the progression rate. The longer you leave the reveal the more chance that the initial seed is forgotten but a long burn drop can have the opposite effect too.
Imagine dropping the story of Danien the doppleganger in the first session and then ten sessions in the players find his trail. They hit the Kessian spice mine and find a near dead mine foreman encased in a chitinous substance. He tells them that he hired a new hand called Danien a week before and as soon as they were alone in the mine he attacked. The players will be talking about how that is familiar. Your note taking player will find it in his notes and the hunt will escalate.
The other side of things is that you may drop in clues from the other subplots at different rates. It does not need to be a long-term thing. On the hunt for the pirate captain’s hidden council chambers, they find reference to a payment to the assassin Nightgaunt against the bartender. They then shake down an informant to find the location of the hidden starbase and he thinks they are there to find out how to contact Nightgaunt. It can be a constant reminder but never allow the plot to come to fruition unless they chase it.
Subplots in their background stories?
Always try to get the player to offer up some background to their character. Treat this as a special thing too and try to build subplot material from it. Perhaps they mention a special member of their family or a special friend. Surround a subplot about them and have the character hear of it. I used to, when I was a player a lot, make characters who were orphans because everyone in the family died. That saved the GM capturing people in the family as leverage. Well, if I had a player do that now then I would weave the deaths into a subplot. Perhaps the family were actually running an illegal drug ring and were killed by the local warlord for failing to please him. Building this story up around the player as they track down whatever the group is pursuing.
This attention will make your game come alive
Many games, mine included, can get too focused on one plot. Everything is designed to involve that plot. Having subplots included causes the game to come alive. Suddenly there are stories in places that the players are not looking. Non-player character’s (NPC) are talking about things that are not directly related to the task they are on. It may have an effect on the plot but it is distinctly different. These touches make the players feel that the world is a much bigger place. They have options to play around with when the main plot is over. In fact, a well-seeded subplot may feel like they are instigating a change. All of this adds beauty and richness to your game universe.
Subplots do not need to be majorly mapped out. A direction on where it is going and some notes about how this information can be revealed. A small notebook and a few dot points can really bring your game to life. Give them a try and keep rolling!