It's a PCV system (Positive Crankcase Ventilation).

Basically, when your engine is spinning, you get peaks of high and low pressure in the bottom end of your engine. This is bad - if the pistons are having to fight positive pressure in the bottom end, it robs you of power. The answer is just to open up the bottom end (ie, don't seal it air-tight) so that the pressure doesn't build up. The hole on your valve cover is to vent the bottom end of the engine. Engines have had ventilated crankcases for almost all of history, but up until the 60s/70s, they'd always just be vented into the air with a breather cap.

So why does it go to your intake now? Well you can't vent that into the air any more. There's all kinds of nasty oil fumes and spent crap floating around in there. Very bad for pollution. So for the last 3 decades, the government has required manufacturers to vent all those fumes back into the engine. There it gets burned cleanly.

If you have a poorly designed PCV system (like my POS Mustang), or a bad PCV valve, it can suck oil into the intake, though that's not generally a huge problem on Hondas. Most people remove the PCV valve either (a) for looks, or (b) for engine 'cleanliness'. Passing all those nasty oil fumes into the chambers to be burned hurts power a bit (I must stress a BIT - might rob you of 0.5hp, really not worth worrying). You're fine removing them, just be aware that depending on your local rules, you may or may not pass emissions, and your conscience may have issues with venting all those bad fumes into the air.

I must also stress one other point - do not, repeat, DO NOT put a breather on the intake side of the tube. Put a breather on the valve cover, but solidly PLUG the intake tube. You can do some serious damage from a lean or rich condidtion if that tube isn't air-tight downstream and around the O2 sensor.