(part of common anime words)
The word ikemen is like the word bishounen 美少年 or biseinen 美青年 in that it talks about "beautiful boys" or "beautiful men." Except, unlike those words, ikemen is a kind of slang that tends to the "cool," "sexy," or "hot" meaning rather than the just "beautiful"
Its origins are complicated. Ikemen can be written as either イケメン or イケ面, and it is an abbreviation of iketeru men イケてるメン or イケてる面. A slang coined around year 2000.
The men メン word written in katakana カタカナ comes from the word "man" in English. Meanwhile, the men 面, written in kanji, is a word that means "face" in Japanese. (list of body parts)
The iketeru イケてる verb itself is a slang which means "to look cool," "to look pretty" or "to look hot." That is, "to be lookin' good".
So, after joining these words, ikemen would mean "men who look pretty" or "men with a pretty face." Or "men with hot face" or any possible combination of those. Of course, you shouldn't translate things literally.
The meaning of ikemen is obviously "hot guys," the popular-with-girls type, and there's no point discussing it further.
Ten years after the word ikemen was born to talk about hot guys' hotness and Japan got used enough to the word that even elders started using it, a new word, ikumen イクメン, came to be.
If ikemen is "men who ike" then ikumen is "men who iku". Just like the ike in ikemen, the iku in ikumen is an abbreviation. This time, of ikuji 育児, which means "childcare"
So, as probably nobody could have ever imagined, ikumen, a word based on another word which means "hot guys," actually means "men who want to raise children." To have children, to father children, to become fathers... of children!
Yeah, think about it. Ten years is a long time. Enough for girls in search of ikemen to become women in search of ikumen.
What happens, actually, is that Japan has trouble having children. Its population has come to a halt and with that so has its workforce. For the government and economy, this kind of situation kind of sucks because you'll have less young people working and more old people retired as time goes. Also, they have to
import get immigrants to work in Japan because there aren't enough Japanese people.
So, for both women wanting to have children and for the country of Japan the ikumen are extremely important.