If you are complacent, it means you do not see any danger ahead, you think that all obstacles have been overcome, and therefore you relax and do not try as hard as you had been trying.

An example of complacency would be the hare, in Æsop's fable of the hare and the tortoise. The hare is so certain he can win that he decides he can lie down and take a nap, and the tortoise beats him. The hare was too complacent.

"There is no room for X" is actually a curious idiom, and I'm not sure that I can explain it, except to say it means "X is unacceptable," "We must not be X." 

"There is no room for complacency" means "We must not be complacent." 

Perhaps the idea behind "no room" is "we have only so much room in our group, we do not want anyone who is complacent taking up space, if you are complacent please go away so we can replace you with someone who is not complacent."

During a political campaign, it would be common for someone on the side that seems to be winning to say "There is no room for complacency.

"There is no room for complacency" means you should not let yourself become complacent, or allow complacency to establish itself.  You should stay alert and fully focused on the task.