What is the exact meaning of the phrase "יישר כחך", and what is the gramatically-correct way to pronounce it?
You can see from the following Talmudic excerpt (Shabbat 87a) that the original use was to validate Moshe Rabeinu's action. It would seem to me that current usage is quite the same. When one performs a Mitzvah those around him will confirm that the action was proper and worthy of validation and strengthening (the action or possibly the performer.)
For it was taught, Three things did Moses do of his own understanding, and the Holy One, blessed be He, gave His approval: he added one day of his own understanding, he separated himself from his wife, and he broke the Tables. ... 'He broke the Tables': ... And how do we know that the Holy One, blessed be He, gave His approval? Because it is said, which thou brakest, and Resh Lakish interpreted this: All strength to thee (=יישר כחך) that thou brakest it.
What a great question! Shkoiyech!
Literally it means "He (i.e. Hashem) should straighten your energy." It means that Hashem should guide you in choosing actions that will allow your energy to flow on a straight path from its source on high down to you.
It is correctly pronounced "Yi/Ya/sher Ko/cha/cha". Its Yiddish pronunciation is "Ya/shi/koi/yach".
Hebrew Wikipedia entry says that "ישר" is an Aramaic root meaning "חזק" and that the phrase therefore means "may your power be strengthened" or, for a pretty good English idiomatic equivalent, "more power to you".
As referenced above - Rashi - in the concluding Rashi of The Torah quotes maseches "Shabbos" (87a) where HaShem says to Moshe - "יישר כוחך ששברת". Grammatically, the phrase would be pronounced as "Yishar Kochacha" - may your strength be made straight. In the Hoshanos in the Artscroll Siddur, the word is written as "Yishar" - see page 738. Also, HaShem wouldn't be referring to Himself in the third-person by saying "may He straighten your strength" - referring to Himself. That's why it doesn't make sense to pronounce it "Yiyasher Kochacha.
Most people (wrongly) pronounce it "yashar koach" or "shkoyach" and this is the way it is said in Yeshivish circles and in Yiddish. However it should properly be pronounced something like "yishar kochacha" based on the way it's spelled. I often tell people it's meaning is something like "more power to you" and although this isn't the most accurate translation it gets the point across. "Shkoyach" is used somewhat like "thank you".