Why are circuits drawn like this(see picture) and not closed?
It's a simplification and Vin and Vout are simply references. Basically they're probe points. It's used in order to determine transfer functions of systems. Transfer functions are useful because the resulting equation that you get out of them can give you an idea of how it will impact a signal propagating through them.

A transfer function is H(x) = Vout/Vin. This is useful if you're messing with frequency and AC systems because then you can use frequency transforms to get a system into the frequency domain and then each stage of say a filter can be easily understood. In this way, you can build and understand how a multi-stage filter would act upon a signal/system.

In this case for what I'm assuming is homework or school work, they're just trying to get you used to the format. Basically, with this notation, they'd ask you how does Vout look given Vin where Vin is a variable.

To analyze them, in your head (or on paper) sweep Vin from -infinite to 0 to infinite and determine at what various inputs Vout would transition. For the first case, determine at what points the diodes would turn on and how that would limit the output. In the second case, determine the same thing: when the diodes would turn on and how that impacts the output.