The word 'force' is right there in the question, so we will certainly start with Newton's 2nd law. But first, let's look at a couple of simpler systems before we try tackling this double pulley.
We need to be sure we know how to deal with tension forces. So, let's look at the simplest possible example: a mass hanging by a chord.
Our intuition tells us that the tension force is equal in magnitude and opposite in direction to the force of gravity. To be explicit, let's write down Newton's 2nd Law for this trivial system anyway.
While this may not have been necessary, it's good to get in the habit of writing out Newton's 2nd Law. As expected, we see the the magnitude of the tension force is equal to the gravitational force. It shouldn't come as a surprise, then, that if T > mg then the mass would be accelerating upwards and if T < mg, it would be accelerating downward. If this is not obvious to you, take a minute to work it out for yourself.
We can now turn to a slightly more interesting system. An Atwood machine consists of two masses suspended from a chord that is wound around a freely-rotating pulley. For now, we will assume that the pulley is massless and frictionless.
Before we try to find the equations of motion for the masses, let's consider how we expect the system to behave. First, it is clear that if the two masses are equal, then there will be no acceleration. And if one of the masses is greater than the other, the greater mass will accelerate downwards while the lesser mass will accelerate upwards. More importantly, because we are assuming the chord is ideal and won't stretch, we can see that the magnitudes of the accelerations will be equal. Moreover, the tension in the chord will be uniform (a consequence of the frictionless pulley).
Now, to verify our intuition, let's start our familiar process of writing down Newton's Laws.
Note that I utilized the constraint that the accelerations will be equal and opposite.This puts the equations in a convenient form, because by adding them together (in other words, adding the two left hand sides of the equation and the two right hand sides) the tension term disappears.
We see that our intuition was correct: if the masses are equal, there will be no acceleration, and if the masses are different, the larger mass will accelerate downward.
Now that we have seen some basic examples involving tension and pulleys, we are in good shape to attack the original problem. The first order of business is considering the role that the rope tension plays in this system. We are assuming that the pulleys are massless and frictionless, so we know that the tension everywhere in the rope must be the same --- keep in mind that if that wasn't the case, then different sections of the rope would be accelerating at different rates, and you would wave stretching and/or bunching of the rope, which doesn't happen.
With that in mind, we immediately see that the force pulling on the rope, F, must be equal to the tension in the rope, let's call it T. The lower pulley is being held up by two sections of rope, both with tension T for a total of 2T --- which must equal at least mg to hold up the weight. Therefore, the minimum force required to hold up the weight is mg/2.
