Air sealing, is simply caulking any gaps & cracks or filling them with expanding foam that can leak air. An example, is caulking around the place your garden faucet plumbing penetrates to the outside. Or, plugging the holes in your attic where the wires run from your lower floors into your attic. The big ones I found were where my plumbing vent stacks penetrated the floor of my attic. I had a one foot square cut out of my attic floor to fit like a 6" round pipe. Also, my chimney penetration had about a 2' wider opening than was needed. I closed it off to within 2" and followed their guide using metal. BTW, insulation is usually bad at air sealing, you need something solid like the spray foam, or caulk, or huge gaps I seal with drywall best I can and fill in the remaining cracks with expanding foam or repeatedly spray expanding foam to close them.
Warm air can hold a lot of moisture, cold air can not. In winter, your house air is humid, any air leaks causes your warm humid house air to escape and as it does it cools down and is unable to hold the same amount of moisture, so the water condenses out of it. The more you insulate the less moisture your house can handle escaping. After insulating without air sealing, you run the risk more moisture is condensing than can work it's way through the insulation causing a buildup of moisture and the health problems that creates, also moist insulation is not as effective as dry, and air movement through insulation reduces its effectiveness.
The normal steps that I used was
www.energystar.gov/ia/home_improvement/home_sealing/DIY_COLOR_100_dpi.pdf and don't let the first picture fool you, those people are just way too happy about sealing! Air sealing takes years... I keep finding new things I need to caulk, etc. or wire penetrations I missed. I recommend you get a can of expanding foam, it's great at sealing because it expands so much you're almost garunteed to close any leaks with it and you can seal up pretty large gaps. But, the stuff is unimaginably sticky, don't use it in the living area, the stuff turns flourescent orange over time, and difficult to use because it has to be upside down to dispense, and for the most part one time use so, if you start don't stop until your can is empty. You can go about 30 minutes without using it, then you better waste some else it starts to seal itself in the can. But, it does a great job of sealing air gaps.