Few people translate games by themselves these days (not that it was common back when) but there are generally 3 main jobs:
rom hacker: Many seem to focus on using game specific tools, I am not entirely convinced such a thing is always necessary but people are welcome to work how they will.
Usual way:
For the DS at least follow release threads/import reviews/release dates.
Requests do get looked at for the same reasons.
Sometimes a hacker will pull apart a rom, make some basic changes, gather screenshots and go looking for a translator.
Find rom.
All DS roms use the nitrofile system at the base level (more than a few stack their own on top if it and practically all them have their own custom formats for something) so pull it apart or parse it with something like crystaltile2.
Figure out how the game stores text: can be in file, can be in binary, in the case of puzzle/simple games can be in a picture, can be in a sub archive (NARC and the like) as well as figuring out any pointers that have been used. Sidenote: while the nitroSDK exists and provides enough base formats to make just about any game only a fool would assume that is all they will see.
Once found and the format figured out document the format. Here is where a tool would be made for it in some cases or just converted into a more manageable/common format (I will often stick pointers in a spreadsheet as the manipulation tools there are good enough when combined with other things).
translators: If the average rom hacker spends a great deal of their time learning to pull apart code there is probably not going to be a lot of time to learn a language well enough to deal with the nuances* (slang, wordplay, jokes, use of text to convey something (think the translations of phoenix wright). By similar token the little bits and pieces the rom hacker picks up are probably the same sort of thing as learning all the little bits of a language.
The basics will probably be part of it as it can change what happens at a code level; as a start Latin derived languages aka most of the European languages have about 70 characters for the lot of them (upper case, lower case, Arabic numbers, punctuation, alternate/special characters not in English çêåë....) and it can usually be squeezed into 128 characters aka each character can be defined in 8 bits. 70 just about covers the kana (kanji are more readily used in modern games): you are going to need 16 bits for them almost always).
*reason number 3 why you never use a machine translator.
clean up: Often overlooked or performed by one of the team. In here can also be all the donkey work for the rom hacking side of it.... once you have done a few repointings or divined a few tables or made a few cheats it gets kind of boring and most people go looking for a challenge. This can leave little things like the extra punctuation to be found; you might see it around here with some of the more experienced rom hackers who will come in and give a hand with a few things leaving the rookie hacker to finish up or do the bulk of the hacking work.
Same with languages: you have effectively just translated an essay length of text and as such your grammar and punctuation may have taken a hit towards the end.
If you are new to this I instead suggest hacking an existing rom (new super mario brothers is quite good for this) or trying to figure out how an existing hack was done.