Did the large number of homophones emerge through use of the written language? How do they manage to talk to each other, then? (Honest questions.)
No. Modern Japanese is the standardization of a 3 different scripts. Kanji, which are Chinese characters, katakana, which is a phonetic script made by simplifying Chinese, and hiragana. Katakana and kanji were used for official writing, and hiragana was used for informal writing. The Japanese government standardized the three scripts in 1915-ish and now they're all used together for different things.^
2. How do they manage to talk to each other, then?
Context, context, context. You can tell which homophone means what depending on the context and part of a sentence a word falls in. Sentences frequently consist of just the verb. Where we might say "turn it on" a Jap would say "on". Also Japanese people rarely interact with people outside their circle of friends/business acquaintances, and when they do it's in extremely formal circumstances where everybody knows what to say.
^Generally, katakana = foreign words, kanji = nouns/verbs, hiragana = grammatical markers(particles & verb conjugations)
My favorite example of this was when I was watching "Bushi no ichibun," a samurai movie that came out a while ago. A man and a woman were in a house, and my wife and I could not figure out their relation. The woman says "Hey, you." And we both realized that the characters were married.
"Is modern Japanese so different?"
Yes. Yes it is. Japanese has gone through so many reforms that's it's almost a completely different language. Also take into account that the Take of Genji was the first written novel and you'll see why modern speakers would not be able to read Tale of Genji in it's original format.
I'm sure there are some sentences that require clarification, but this is true in english as well. But generally speaking, it is not an issue. Only very rarely do japanese people end up needing to disambiguate through pantomiming the strokes of kanji in the air or saying "you know, the first character from such and such, not the character from that other word" (except when discussing the writing of names, which is totally insane and is the real life version of the old monty python Throatwarbler Mangrove sketch).
（i) words derived from Chinese (kango) are a large part of written Japanese vocabulary;
(ii) kango came to Japanese largely through the writing system (not through massive Chinese immigration);
(iii) and most of the problematic homophones are kango. There aren't that many homophones in "yamato kotoba", the "native" Japanese words.
So, yes, a large number of homophones emerged through use of the written language. Of course they're part of the spoken language now, but the written language is significantly more formal and so still contains more kango, and therefore more homophones.
For some evidence on (i) and (ii) see http://en.wikipedia.org/wiki/Sino-Japanese_vocabulary
There are at least two reasons for (iii): spoken language naturally leads to different words having different sounds; on the other hand, many words that sound different in old Chinese end up sounding the same when adapted to the Japanese phonology. This is because Chinese has tones and consonants and vowels that don't exist in Japanese. So Chinese "kang1", "kang2", "kan1", "kan2", etc, all become "kan" in Japanese.
As blintson said, context is enough most of the time in conversation. Example for "yamato kotoba": hana can be "nose" or "flower", hashi can mean "bridge", "edge" or "chopsticks". As you can imagine, most of the time the intended meaning will be clear.
From the top of my head, I can only come up with 2 pairs of homophones that often elicit clarification in conversation: shiritsu can mean both "municipal" (市立） and "private"（私立). Annoying when you're talking about schools. Another is kagaku which can mean both "science" (科学） or "chemistry" (化学).
Those words are actually spoken differently, even though they share the same spelling in romanji (or hiragana). The stress accent is in a different place. This is similar to how in English, the noun "record" and the verb "record" are differently stressed despite their identical spelling.
Possibly a phonetic othography that included accent pitch would maintain greater readability once people were used to it. Unfortunately, people in Kansai put the stress on different syllables than the people of the Tokyo metropolitan area.