It's not the temperature, as they say: it's the obliquity. And yet, the connection with winter weather and decreasing temperatures is very real.
At this point, you may be wondering what on Earth I am writing about, and in fact, you would be very close to answering your own question. Our planet Earth is tilted, or rather, the Earth's axis of rotation is tilted with respect to the ecliptic, and the degree of tilt is referred to as obliquity. If you're wondering what that means, it means that an imaginary line drawn between the north and south poles would make an angle of 23.4 degrees (actually, 23.43696 degrees) with an imaginary plane that describes the Earth's orbit. But these imaginary constructs have very real consequences. Only a tilted planet has seasons, and seasons, while not a prerequisite for life, tend to moderate planetary climates and homogenize, so to speak, climatic extremes. At the very least, seasons make life on Earth more widespread and diverse. Right now, the south-pole end of the Earth's axis points towards the Sun. In three months, the axis will point along a line tangent to the Earth's orbit. In six months, the north-pole end of the axis will tilt right at the Sun, and so forth.
The Earth got tilted fairly early on in its formation, probably as the result of some proto-planet slamming into it, disturbing its rotation and its orbit as well. This type of event almost certainly occurred more than once, and one of these collusions was so violent that a large chunk of our planet was detached, and eventually became our moon. (These family histories get complicated.) Changes in the direction of the axis, and changes in the earth's orbit, have resulted in the angle of obliquity varying between 22.1 degrees and 24.5 degrees every 41,000 years. Right now, the angle of obliquity is getting smaller (hence the title.) We're about halfway through the downward cycle, according to my rough calculations.
On the winter solstice, the northern part of the earth points directly outward from the sun. It's the day we in the northern hemisphere get the least amount of daylight. It's very seldom, if ever, the coldest day of the year, because it takes some time for that lack of sunlight to have its maximum effect on our weather, but it traditionally marks the real start of the winter season. The coldest days will come later. The good news, for plants, for animals, for gardeners, and for human beings in general, is that every day for the next six months will be longer and lighter. Happy solstice!