Infantry is the cheapest and weakest unit. But it is also the most often bought unit, and one cannot hope to win without using infantries. And in the first few turns, it's all you can afford anyway :)
Infantries have low attack power: they are good against other infantries and mechs, not very good against transports and ranged units, and cause only scratches to anything else. But they make up for firepower by their cheapness and versatility.
Infantries are the only unit (with mechs) that can capture properties. Infantries are also very useful as meatshields, typically by protecting artilleries, because they can often take two attacks before dying. They can also be used to restrict the movement of the opposing units (yeah, even air units).
In AWBW, having the most powerful units does not always mean victory, as multiple cheap units can win against a lone powerful unit. Total unit count is quite important, and building infantries are a mean to achieve this. As a rule of thumb every turn, rather than buying just one big unit, it's better to at least buy infantries on bases where you can't afford more.