Why does the sun feel hot through a window even when it's very cold outside?

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It has to do with reflection and how glass magnifies heat. When the sun streams toward the window, the room or person near the window gets hot. Yet, if you touch the glass, the glass itself may feel cool or cold because the colder outside temperature affects the surface of the glass.

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More specifically, it's because glass lets the energy in as light, but once it becomes heat, the glass doesn't let the heat back out. The energy gets trapped inside the building as heat.

And another answer:
More specifically still, heat can be transported in three ways, known as convection, radiation and conduction.
Convection is when a fluid (could be water, could be air) gets warm, moves to somewhere else and gives up the heat it got. This doesn't apply to this question because the window won't let air pass through it.
Radiation is when a warm object (could be the sun, could be a red-hot poker) gives off radiation. This radiation can easily pass through a window.
Conduction is when an object is warmed in one place the heat travels through the object and arrives somewhere else.
A glass window is good at transmitting radiation - that's what transparent means.
Glass is a rather poor conductor of heat by comparison. The glass does not magnify the heat in any way : all it does is to allow the sun's rays to come in with almost no hindrance, while slowing down the loss of heat to the cold air outside.
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