Time is of the essence means it is critically important for the reader or hearer of this phrase to act. This phrase puts on notice the recipient, that failure to act is likely to make the recipient liable for negligence following any consequential loss. Anyone who uses such a phrase is doing two things
Communicating the importance of the matter under discussion and idiomatically
covering their own back to pass the buck onto the next person in the chain of communication so as to avoid consequential personal liability. If anyone is going to be responsible it is not going to be the person making this statement! Be warned!