How do 'greenhouse gases' let heat in, but not let it out? That honestly doesn't make any sense to me. How could heat pass through a gas one way but not the other? Its not like our upper atmostphere has a bunch of doors that can only open one way. To me, that seems to fly in the face of everything I've been taught about how the physical universe works.

If heat can pass through a substance one way, there's no reason why it can't go through the other way. Except for 'greenhouse gases' apparently. 

The closest thing I know of that works this way is one-way mirrors. But that doesn't have the same effect. The reason you can see through one side and not the other is because most light is reflected, but not all. Thus if you're in a dark room, the light coming in from the outside will overwhelm what little light is being reflected from within the room. But if you're on the outside, virtually no light is coming through from the dark side and thus you can only see what is being reflected.

In short, it makes no sense to me and flies in the face of everything I've been taught. Now, I'm not a climate change denier. Actually, this is something I just now thought about. I've never really bothered to question it before.
In a nutshell:

The radiation that enters is shortwave radiation from the sun.  Solar radiation is dominated by visible (as well as UV and near infrared) radiation with a wavelength mostly between 0.2 µm and 2 µm.  This wavelength is determined by the temperature of the Sun, in the order of 6000 K.  For visible radiation (roughly between 0.4 µm and 0.7 µm), all the gases in the atmosphere are (almost completely) transparent.  Clouds do scatter this radiation so you can't see the Sun directly when it's cloudy, but they still let plenty of light through, so it's still light on a cloudy day (just less bright), although it does feel cooler.

The radiation that leaves is longwave radiation, also known as terrestrial radiation or (incorrectly) as thermal radiation.  For Earth, this is dominated by radiation with a wavelength of mostly between 4 µm and 40 µm.  This wavelength is determined by the temperature of the Earth, at around 290 K.  At these wavelengths, greenhouse gases are mostly opaque.  They absorb the radiation.  The greenhouse gases reradiate the absorbed heat, but in both directions (back to Earth AND up to higher layers in the atmosphere), and at a lower temperature.  The upward longwave radiation gets in turn absorbed by higher-up layers of greenhouse gases, et cetera, such that ultimately the layers of greenhouse gases that do radiate into space tend to be high up in the air (say at 240 K).  Thus, their overall effect is to radiate less heat into space than the Earth surface would in the absence of those gases.

The most important greenhouse gases are water vapour, carbon dioxide, and methane.


Source: gisgeography.com  


Source: science of doom