How can I fix stuttering motors that are connected to ESC's [Arduino Uno]?
It isn't clear from your diagram or the video what wire thicknesses you've used or precisely how currents are being routed, or how big your battery is.

Let BPG denote the ground terminal on the battery pack, and BP+ the battery's positive terminal.

There should be heavy wire from BPG to the ground pin of each ESC.  There should be a signal wire (that is, heavy or light, makes little difference as long as motor currents have their own heavy wires to follow) from BPG to an Arduino Gnd pin. 

There should be heavy wire from BP+ to the positive voltage pin of each ESC.  There should be a signal wire from BP+ to Arduino Vin.  

There should be signal wires from Arduino PWM outputs to ESC PWM inputs.  There should be heavy wires from ESCs to motors.

Also, in your code try changing 100 on up to 200 or 250 (or near whatever the ESC maximum setting is) to see if problem persists at higher levels of PWM drive.  Also if your battery pack is small, add some more packs of the same voltage in parallel, or try a heavier-duty pack. 

Edit 1: Check whether each of the motors can work ok by itself.  That is, either comment out three at a time, or use if-statements that check button or jumper settings.  If all the motors work ok singly, try them in pairs or triples.  There are six different combinations of motor pairs, so rather than trying all six of those you might first try the four different sets of three motors at a time, and if that reveals a combination of three motors that fails, try the pair combinations among those three to see if there is some interaction between two motors.  If problems occur when testing with fewer than four motors, try swapping motors on to different ESCs, to find out if the problem is due to a motor, an ESC, or the battery.  (You can also swap motors on ESCs and test all four at once, but testing with fewer motors going should provide a more systematic problem solving method.)