The issue: Your startup disk is full.

Your startup disk is full.

To make more space available on your startup disk, delete some files. If you’re sure that your disk has enough space and the space reappears after a reboot – most likely cause is you don’t have enough RAM, have to many Applications open at the same time or one of your applications has a memory leak. Try updating all software and using Activity Monitor to figure out which process is eating up the disk space.

Another possible culprit is Time Machine:

Time Machine includes a feature called “local snapshots” in OS X Lion and later. This feature keeps copies of files you create, modify or delete on your internal disk when your backup drive is not available.
Checking free disk space
You can see how much space is being used by local snapshots from the “About This Mac” window.

Select About This Mac from the Apple menu.
Click the More Info button in the window that appears.
Click the Storage tab to see available and used disk space.
The space used by local snapshots is labeled as Backups.

A recent post on Apple support communities, titled, “Guide: How to solve Yosemite memory leaks and CPU usage” looks promising as a possible solution to this issue:

I post this hoping to help those users who, like me, are experiencing high CPU usage and massive memory leaking with OS X core services and apps, leading to slow performance and battery drain.

 

I’ve tried everything mentioned, but found the right combination of steps to follow. I’ve tried this with seven different Apple computers, including mine, and has worked well so far. I applied this method yesterday to give these process a 24 hour window to fail again, so far everything good.

 

First step: disconnect any external or secondary monitors, if any is present. The video memory allocation leak can also happen if you have a system with an integrated card, like Intel, with no external monitors attached.

 

Second step: Shut down your machine and enter Safe Mode (press shift once you turn on your computer again, more info below). Once you’re there, fix your disk permissions.

 

Guide of how to access Safe Mode: OS X: What is Safe Boot, Safe Mode?

 

Third step: From Safe Mode turn off your machine again and reset your System Management Controller (SMC). There are different methods, depending on machine, to do this. To know what method applies to yours read the following guide.

 

Intel-based Macs: Resetting the System Management Controller (SMC)

 

Fourth step: Once your machine completes a full boot after resetting the SMC turn it off again and reset your PRAM (THIS IS THE MOST IMPORTANT STEP, BUT THE PREVIOUS ONES ARE ESSENTIAL FOR THIS ONE TO WORK). The PRAM stores small bits of data that indicate our Apple computer how to interact with the devices connected to it, including monitors and video cards. It also affects software.

 

To reset it you should hold the OPTION, COMMAND, P and R keys in your keyboard immediately after turning on your machine again.You’ll hear the start up chime, continue pressing the keys until the machine boots and the chime starts A SECOND TIME, then release. IF YOUR YOSEMITE INSTALLATION LOCKS UP AT A BLACK SCREEN AFTER THIS, DONT PANIC! It’s normal, just turn off your computer and let it boot again.

 

More info about PRAM: OS X Mavericks: Reset your computer’s PRAM

 

Voila, reconnect your external displays and enjoy your system.