I know other words like 如果 also mean "if", but why would 要是...的话 mean "if"?
An example sentence:
If you like Chinese, you should listen to Chinese music.
If you find these kinds of things puzzling, I suggest you try and download 现代汉语八百词. It gives meanings and uses of a lot of these constructions. As Huang says, these constructions mean just what they mean.
On page 594 of 现代汉语八百词 it says (the page number will depend on the edition you download):
～看见《汉英词典》，替我买一本 | ～他不去，你去吗？
～有人问的话，说我在老马家 ｜ 坐船去好，～来得及的话
c) 要是 ＋ 名。
～别人（＝如果换了别人），这事不一定能办成 | 老同学聚会真不容易，～去年，咱们还聚不齐呢！
Meaning is roughly:
[Conjunction] expresses hypothesis; if.
a) used in first clause.
If you see a "Chinese-English Dictionary", buy one for me | If he doesn’t go, will you go?
b) 的话 can be added after 要是. '要是…的话' can be used in the second clause.
If anyone asks, tell them I’m at Ma’s place | It’s good to go by boat, if you can make it.
c) 要是 + noun. If someone else (= if it were someone else), this matter could not necessarily be accomplished | It’s not easy for old classmates to get together, if [it were] last year, we couldn’t have done it!
Great answers by Terry and Bathrobe.
In this instance, when I bumped into the same thing, I thought of it as the English
In that way "say" gets a nice and logic connection with "话".
Unfortunately, one of the things about languages is that they are "arbitrary". They basically just pick a group of sounds and declare, "This is what that means."
In English, for example, a butterfly is not flying butter, even though the two smaller parts one can see in that word have those meanings. In the same way, it doesn't help much to think that one can break everything in Chinese down and get logical explanations for everything. Language is just that way (otherwise computers would have taken over our function a long time ago!)