So I live in Texas, and it get awfully hot in the summer- like 37°C hot- and yet, when winter rolls around, it can get down to 4° or 5° pretty easily. So, my limited understanding or weather and stuff doesn't understand one thing. Where'd all that heat go? It was so hot in summer but it'll disappear in a week. I know cold is just the absence of heat, so the heat must've gone somewhere, right?
The heat goes away in two main ways. The first is radiation as mentioned in a comment. Heat radiates away into space or it may warm the air before it gets to space. That leads to the second reason heat goes away - convection. Weather systems move air from one place to another. The hot air literally blows away or rises and is replaced by cooler air that blows in. Where there is mixing, the warm air warms up the cooler air so the mixed air approaches the same temperature.
There are a couple of other ways that heat is lost. First, water in lakes and rivers evaporates so you get evaporative cooling like a swamp cooler. That is most noticeable in summer but happens at a slower rate as air temperature decreases and as the sun stays up for fewer hours in the autumn and stays lower in the sky. Plants transpire water vapor from their leaves and that has the same effect as evaporation. The other way is by melting snow or ice. When snow falls on warm ground early in the winter, the ground has retained enough heat to melt the snow. This cools the ground to the point where the snow will stick.
User 32214, I suggest you draw back from Texas, disregard local weather anomalies (which don't count for much), and look at the bigger picture. Major seasonal changes in the heat distribution occur by three main processes. By far the most important is day length / relative accumulation of solar energy caused by the way the Earth's spin is inclined to the plane of orbit of the planet. This yields the normal seasonal variation. In addition, Hadley cells redistribute tropical heat polewards, whilst continental-scale (and larger) loops in the various jet streams bring thermal anomalies on a scale of weeks or even months. These processes bring inter-annual variations in the climate at any given location. Jet stream meteorology is rather complex, but a good summary is found in the clip: https://www.youtube.com/watch?v=6aLk6b0vNUc The effect of climate change is to push extra heat polewards. This reduces the thermal gradient between either side of the polar jet stream, thus making the jet more sluggish in its migration around the world. This sluggishness causes the jet to slow, and create north-south loops, yielding extra cold anomalies on the cold side, and extra hot anomalies on the equatorial side. These account for record breaking hot and cold anomalies in North India, Moscow, Western Europe, and maybe Texas (though I am not familiar with Texan weather systems). So the short answer to your question is in two parts. Firstly, there is the major annual shift in climate due to the Earth's spin angle relative to the sun. Second, less important, but more interesting is the that climate change is very slightly beginning to alter the heat distribution of the planet. Or, you can adopt Donald Trump's 'science-free' head in the sand policy, in which case any climatic trend is a total mystery!
Recently we have seen a rather odd scenario developing, in which the polar and subtropical jet streams have been merging. These are normally very different and distinct. Maybe this is a one-off anomaly, or maybe we are entering uncharted territory, where it will be harder to predict hot and cold spells - at least until we get a better handle upon how climate change will influence future jet streams.
PS another factor is the annual oscillation of the inter-tropical convergence zone, but I suspect Texas is too far north to be involved in that process.