If a "literal" translation doesn't make sense to you in the first couple of seconds, then forget about it - Irish isn't just English with different words, it's a completely different language that predates English, and English picked up lots of usages from other languages that aren't reflected in Irish usage. Relying on "literal translations" to explain things to you is a bad habit if you expect it to always work.
In this case, the "west" meaning comes from original "behind" meaning, not the other way around.
I can't speak for anyone else, but I'm pretty sure it isn't a coincidence that the words for the "left" and "right" happen to align with the words for "north" and "south" in Irish, and that "behind" and "west" are the same. I'm sure someone has written a paper on it, but I haven't looked.
Even in English, it is believed that "East" derives the Indo-European root word for "shine" and that the word "Orient" (as in "the Orient") is derived from the Latin verb for "to rise". Irish hasn't gone through as many mutations as English has, so it's easier to see the connections between the words.
The s in tusa is a broad s (sounds like "ss"). The s in Tá sé is a slender s (sounds like "sh").
A slender s and a broad s are two completely different sounds, and it is clearly a slender s in this recording - Tá sé, not tusa.
You can access the audio directly here, and slow it down to hear it more clearly.