Step 1. For various reasons the normal versions of windows that are windows XP SP2 or later are terrible for various methods of this (though you should be able to get some stuff captured on windows). Server versions should do OK though for some of this though. Linux and BSD should be fine. I am not sure what goes with Macs. Linux runs great off a liveCD though and there are many network focused and even network hacking focused distributions. Virtual machine versions of linux with windows as a host may or may not trouble you depending upon the network setup.
Short version is if you are not running linux/BSD then get a liveCD until you can argue otherwise or are just interested in dumping, you will also have a hard time if you do not have much spare network gear sitting around. As most of you have linux devices in the form of android whatevers which tend to have wifi you might have some options compared to doing this a few years back.
Back on topic there are two things people are seemingly wanting to do
1) Just analyse the data sent in.
2) Change the data.
Analysis.
For this you need to get the packets on your computer. There are five main methods
1) Own the router/run your own. Here you get a spare wireless card, some method of internet access that does not involve that wireless card and a computer upon which to set up a hotspot. As you own the computer you can monitor what goes through it and onto its various ports. Some have some luck with tethered android devices.
2) Own a bloody expensive router (talking proper Cisco/Juniper expensive) that supports various techniques in the multicasting world and/or forwarding of network frames (various terms are used by various vendors I believe). To save most of you the trouble you do not own this sort of hardware and if you do then you probably already know all this, in the consumer world I do not think even something like DD-wrt hacked devices can pull this off convincingly though I am prepared to be proven wrong there. Main exception is if you built your own router (
http://linitx.com/product/adsl2-pci-card-single-port-adsl-modem-annex-a/12181 ) though most people that would do that know all this and more.
3) Depending upon the type of network security (read not if you are using WPA2) you have you can probably use Monitor mode or Promiscuous mode of your wireless card. Most typically this will see you use wireshark (though wireshark is generally a great monitoring program and will likely be used by all of these). Later versions of windows again can trouble some of this.
4) ARP spoofing and related techniques. Will be used kind of like 2) actually but more or a hacker way of thinking about things. Here you make it so your computer is in fact the conduit through which everything passes and if everything passes through you then you get to look at it. ARP is the system that governs some of the lower layers of networks (nice picture
http://blog.buttewifi.com/wp-content/uploads/2009/08/arp-poisoning.jpg ). Naturally if you do not know what you are doing (and not many will) you can hose up your network enough that you will need to either flush caches, fix what you did or more likely just reboot things.
5) The more legitimate version of 4) where you set yourself up as a default gateway and then pass things on to the actual internet gateway for your network. Most would probably do this by a)disabling their DHCP server on the router and making one on a system they control to point at a device aimed at routing or b) manually configuring the 3ds to point at the gateway of your choosing (namely the machine with a packet scanner). Depending upon the setup this sort of thing is also one of the ways the lesser sysadmins of the world will run a web page filter though do learn about networks before you go trying to bypass such things at work/school as it is not the only use for a gateway setup like that. It is used more for those using a VPN (
http://tinc-vpn.org/examples/redirect-gateway/ ) but it works the same here.
The best bet for most people is 1), however it will require your own hotspot and so a few pieces of network hardware that can be reasonably configured, if you participated in the streetpass faking stuff that was found back in August and did not change your mac address at the router (quite likely as few consumer routers have this option) you probably have enough gear for this, though there will not be nice scripts and you may to think through it if you are fiddling with raspberry pis or something. However this is not most "given to you by the ISP" level hardware not everybody will have a few spare wireless cards (amusingly this will probably be a "if you understood enough to do it then you probably have the hardware capable of doing 1) available to you" type situation) and beyond that whenever we do things like this we often hear "but everybody else also uses the wifi and if I am playing hacker...". Again Android type things have a few options here.
Once you get your packet (incoming ones should be using the UDP protocol, usually 989 in length, though various things can see the size increase, and aimed at your 3ds mac hence the filter that filters for your mac and lengths of over 900) you get to seemingly slice out the pokemon portion of the raw packet and feed, I have not paid much attention to this part as it also appears some are improving this to take "raw" packets instead. If by some miracle you are doing 2) from above then depending upon the vendor/setup the mac might have changed or it might be encapsulated and no longer 989 in length.
Changing. The technique for it is known as packet injection. It is done in three main ways
1) The best method involves the use of so called Raw sockets. Windows XP did ship with them but they were criticised as only really being useful for hackers (though such a criticism was itself widely criticised as baseless), with the rise of botnets that also used them in the middle of XP's lifetime Microsoft basically stopped them with XP SP2 and this has remained for all non server versions of windows ever since.
http://msdn.microsoft.com/en-us/library/windows/desktop/ms740548(v=vs.85).aspx has more if you want to get technical.
2) Technically the network hardware communications pro specified by Microsoft and 3Com called NDIS (though unless you are a driver programmer you would have probably met it in the Linux world as NDISwrapper instead) does/did have some support for raw commands. Much like raw sockets above Microsoft has nerfed it as far as being useful to hackers goes.
3) Driver level shenanigans. More commonly seen in those doing certain types of wifi hacking, this is why you might have clicked on a link to various types of wifi cracking site reading through threads like this. Unstable as you like on windows (indeed most wifi hacking tools will not support it as it is too much of a headache) and not always that much better on Linux. Will tend to require specific hardware too as not everybody open sources their drivers or has things hacked well enough to have open source drivers made.
Again the short suggestion is just get a version of linux you can play with. As long as it is not completely locked down/application specific (
http://distrowatch.com/dwres.php?resource=major being a nice list of general distributions) you should be able to get it done. You need not get a specialist hacking distro or one with such a package, indeed such things are often a bit scary for those new to linux, but such a thing would come with it.
The two main things to do would either be replays (presumably of pokemon that are good/rare) so as to clone them or straight up modification/creation. Right now it seems neither are possible owing to either network level fun or due to game level encryption though both seem pretty viable.
Not all packet injection software is made equal and though you can inject an arbitrary packet with most of them if you have to edit in real time and can not edit the hex fast enough to avoid a timeout then you are going to need something a bit more special and that means
http://www.secdev.org/projects/scapy/ (a fairly complex tool) or maybe
http://code.google.com/p/ostinato/ if you are lucky, unless you somehow want to chain a dumping tool, an editing tool and an injection tool to do it all for you.
