"Shitsurei shimasu." is Japanese for
"Excuse me but I must be leaving now.". You would use
this when leaving the office to apologise to your
co-workers for your slackness in going
home before them. Your colleagues will probably
reply "otsukare-sama deshita", which means "Thank you
for your hard work".

Like so many phrases in Japanese (or any other language), the meaning of "shitsurei shimasu" is not adequately explained by giving the "equivalent" English phrase for one of its applications. There exists no 1:1 mapping between words in different languages.

In fact, "shitsurei shimasu" can mean the exact opposite of beable's "translation", as it is also used , e.g. when entering someone else's home.

For a broader explanation, let's first look at the literal meaning, which is quite enlightening: shitsurei means "impoliteness", and shimasu is the neutral polite form of the verb to do. A literal translation might be "I'm doing you an impoliteness" (since Japanese quite often depends on context to supply subject or even tense of a sentence).

Basically, the phrase is employed to preemptively seek excuse for small impolitenesses that hardly anyone would feel offended by anyway, such as leaving a meeting early, "invading" someone's home (even though invited), or (as an employee at a shop) briefly obstructing a customer's view of the merchandise while passing in front of them in a narrow alley between the shelves.