Why is air pressure greater at the poles

The amount of solar radiation arriving at the earth's surface and in the atmosphere is much greater at the equator than it is at the poles.
This difference drives a temperature difference, which in turn drives convection cells in the earth's atmosphere, because hotter air expands, becomes less dense and then rises, while colder air contracts, becomes more dense, and then falls.
Heat must flow naturally from the hotter equator to the colder poles, and convection is a major means by which this occurs in the atmosphere.
Continually heated and rising air forms large low pressure areas near the surface. This happens consistently in latitudes near to the equator due to the high surface temperatures. 
Continually cooled and descending air - subsidence, forms large high pressure areas. This happens at the poles, where the temperatures are coldest on earth's surface.
Hot air near the equator, for example on the ocean surface, expands and becomes less dense than colder air directly above and around it, so at the equator, air heated above the warm sea water constantly rises from sea level up to high altitudes, where it is displaced by the flow continuing from below, and moves either North or South. Air at equatorial latitudes (say +10 to -10 degrees latitude) also contains a great deal of water vapour and thus a lot of excess energy, so as it rises it expands and cools and creates lots of clouds and rainfall. The constant upflow of air at the equator is the reason air pressures are generally low in latitudes near the equator.
In the sub-tropical regions (say 20-35 degrees North or South latitude), the warm equatorial air is descending again, having lost a great deal of its water vapor - as this warm dry equatorial air descends it is heated again - this subsidence produces relatively stable sub-tropical high pressures.
The reheated warm air from the sub-tropics in turn rises and flows North or South from 35 degrees latitude; North or South to about 60 degrees from the equator, where it meets very cold air moving to lower latitudes, from the polar regions.This creates fairly consistent low pressure areas, known as the sub-polar lows, at the far north and south of the temperate zones - a lot of precipitation is caused by the weak cyclonic storms that result from these lows.

At the poles themselves, 90 degrees latitude, the air is very cold and dry, there is little heating from the sun, and the air constantly subsides, forming weak but very large cyclonic systems called the polar vortices. The descending air causes high pressures, called the polar highs. The south polar high is stronger than the north because it is over land.
Now the ideal gas law is PV=nRT, n is the number of moles of the gas, and you correctly note that this implies, if V is fixed, that P would be higher for the hotter air at the equator than for the colder air at the poles. But the V of a parcel of air is not fixed, it can be heated and expand, causing it to rise, or it can be cooled and contract, causing it to fall. The mass upflow or down flow of air in a given region is a factor that affects the air pressure.
The density of the atmosphere near the surface is not constant, in other words.
At the poles it is very cold and dense, and the equator it is warm and less dense. The density effect and the mass airflows are what win out in determining the air pressures

Density and pressure