How do I find the right meaning of homonyms 同音異義語 via audio?

Question

Japanese has a lot of homonyms and sometimes one has more than 10 meanings.

When I heard the following sentences on the phone or a radio. How do I deal with it?

1. はしのはしですか？
2. きのう、かきをたべたけどあまりおいしくなかった
3. これはすいせいですか？きんせいですか？
4. かいじょうには、たくさんのえいががありました

Answer 1

1. By accent. See: Is there any difference when pronouncing 橋 and 箸?

   はし【HL】（箸）、はし【LH】（端）、かき【HL】（牡蠣）、かき【LH】（柿）

2. By context.

   • すいせいですか、きんせいですか？　→　水星ですか、金星ですか？
   • すいせいですか、ゆせいですか？　→　水性ですか、油性ですか？
   • すいせいですか、りくせいですか？　→　水棲ですか、陸棲ですか？
   • ぎんせいですか、きんせいですか？　→　銀製ですか、金製ですか？

3. By actually changing the reading for known confusing pairs. See: How to Pronounce 化学 "Chemistry"?

   化学【ばけがく】、市立【いちりつ】、私立【わたくしりつ】

If everything above fails, you have to rephrase or explain it (This rarely happens in practice).

私はほうそうかんけいの仕事をしています。

• あ、つまり、法律の。 → 法曹関係
• えーと、ラッピングの。 → 包装関係
• テレビやラジオ放送のほうそうです。 → 放送関係

Answer 2

I think that good answers were already given, so I was initially writing a comment but it got long hence I decided to post it as an answer. It might be a good reference for every one after all.

I wanted also to say that probably by accent and context is the main way.

In this regard, I wanted to add that not long ago I found this interesting EDX class on Japanese pronunciation from Waseda University.

Despite I have been living in Japan and studying the language for quite a few years now, It was actually the very first time I encountered some proper explanation about pronunciation (but probably this because I have never formally studied in a Japanese school).

Anyway, there are lots of audio material and at least for me, despite I consider myself to have a good ear, it was quite hard to catch some very subtle (and interesting) differences. As I mentioned above, it could be a good reference for Japanese learners to specifically practice pronunciation.

My answer might be a bit a duplicate (sorry about that) but as I said I mostly wanted to add that reference with some explanation, as I believe other readers might be interested.

Answer 3

Since so many stress the difference in pitch accent I decided to post my comment as an answer instead.

Even though it does exist, the pitch difference is not what Japanese native speakers normally use to distinguish words, but rather it is done by context instead. Also, the pitch varies a lot between dialects throughout the country, and in some Tohoku dialects they don't even have it in the language, yet people communicate without any problem with each other (as for dialectal traits, intonation is usually persistent despite attempts to speak 標準語)

Here is an article discussing the (non-)necessity of pitch accent in Japanese: http://www.sciencedirect.com/science/article/pii/S0911604413000547