LINE or whatever else is popular for showing your friends how many friends you have. I mean, in order to get things done in Japan (just like elsewhere), it helps a lot to "know a guy" or to have some connection.
Examples? In all the job interviews I've passed here, the interviewers at some point dropped the phrase, 何か縁があるようですね。 "Oh, you went to that university? We're invested in a research center there." "Oh, you play concert piano? We're affiliated with a conservatory." "Oh, you taught at that school? My daughter's a graduate." Who your family is, where you went to school, and who you work for are all a big deal, and being able to say, "I was referred to you by such-and-such," can open doors that may have otherwise stuck fast. (Interestingly, this is the same country in which references are often unasked for in job applications.)
But perhaps the biggest examples of this reference culture in Japan is the guarantor.
In Japan, a 保証人 (hoshounin) is the person who takes financial and social responsibility for you when you step from childhood (high school, university) into adulthood (full time work), and he'll remain there in the background for several more years as you establish yourself as a stable member of society. A 保証人 might be asked when you take on financial responsibilities, like applying for a credit card or renting an apartment (called a 連帯保証人, rentai hoshounin, in this case), or when you take on new social responsibility, like starting a new job (in which case he'll often be called a 身元保証人, mimoto hoshounin).
Furthermore, the guarantor himself has to meet a certain set of requirements in order to be accepted as one. In traditional Japanese thought, the 保証人 is a middle-aged, ethnically Japanese male with stable employment and a blood relation to the applicant. Even today this is the kind of guy who will pass application screening with flying colors, and most young Japanese people continue to list their fathers as guarantors if possible.
Of course, it's unreasonable to expect every guarantor to fit this cookie-cutter image. Individual banks and landlords make the final decision on which guarantors to accept and which to reject, but the basic requirements of a guarantor are usually:
1) Related to the applicant (parent, sibling, etc.)
2) Having stable income sufficient to recompense for any debt the applicant could incur
3) Being available and easy to get in touch with (having a stable address in Japan, preferably in the same prefecture or region as the applicant)
The applications that receive the closest scrutiny are of course ones that involve taking on debt (a 連帯保証人). The 身元保証人 requirement of new jobs is in my experience much more flexible, and I've been able to list non-Japanese relatives living out of the country. You can read more (in Japanese) about the different legal responsibilities of the 連帯保証人 and 身元保証人 here.
As a final word, recently there has been a flourishing niche market in Japan of companies (保証会社) that will provide a 保証人 service on your apartment rental application in exchange for a fee. That is, you can pay them (usually an up-front sum followed by annual premiums) to sign as a guarantor for you on an application. If you are absolutely strapped for a guarantor, this might be an option to consider. However, your landlord still has the right to accept or decline the guarantor you provide, and they may not be willing to accept a company in place of an individual. On the other hand, you could be asked by a skittish property manager to sign up at one of these 保証会社 companies even after you've provided a traditional guarantor, as was the case for me in one of my previous apartments.