The standard filter is a disposable dry paper element, and it seems that the one you have is a reusable filter which is soaked with oil. This has a few advantages – the first is that if it was not reusable they wouldn’t sell many of these – for the price, owners want to know they don’t have to bin it after a year or two. The second is that the oil attracts dust particles, which allows the filter element to be coarser than the dry paper element, and as such less restrictive.
As for the theory…. well the engine burns a mixture of fuel and air. You can get more power if you burn more mixture in a given time frame. But while you can easily pump more fuel in, how do you get more air in - short of fitting a turbo or super charger?
So you try to improve the flow through the head. Remove restriction on intake by using an air filter with better flow, improve the inlet manifold to have less bends and smoother inner surface, improve the head and valves (bigger valves / higher lift & longer duration cams / etc), upgrade the exhaust manifold to tubular steel again with no sharp bends, and replace the standard muffler with a free flow one (and if you don’t mind about being road legal then bin the cats while you’re at it)..
Oh, and to take advantage of all this you will also need a modified chip which means a custom map designed using a rolling road engine dyno.
The important thing is to upgrade everything at the same time, so that you are removing any bottlenecks along the way. But even so, with this mods alone, you will not see more than 10% power increase – if that… when you are building race engines, every little bit counts.
In that context, just replacing the air filter by itself will have little effect – the standard unit is suitable to the engines characteristics, and if you do not upgrade the rest of the path than apart from a deep roar you will not see much improvement. But if you like it then why not… and if it is made by a reputable manufacturer then I doubt if it will have any adverse effects on the engine. I have been running two different cars on K&N filters for about 10 years in total, had no problems whatsoever.
Hope this help…