Schtum is a slang word meaning 'quiet'. It is most often used n the phrase 'keeping schtum' or 'keep schtum', but is used in other contexts. It is most often used in the sense of keeping one's mouth shut, rather than sneaking around quietly.
It appears to be from the German word stumm, meaning 'silent'. Alternate English spellings include stumm and shtoom; schtum has probably gained its position as the preferred spelling primarily from the fact that it looks like Yiddish; 'shtum' is used in Yiddish to mean 'quiet', but there is little evidence that this is where the British slang term came from. The UK term was apparently brought into popular usage in 1958 by Frank Norman, in his book Bang to Rights: An Account of Prison Life, where he used the spelling shtoom ("I think it's much better to keep shtoom") and shtoomup ("You can always shtoomup if any screws are earholeing").
Schtum is still primarily a UK slang term, but is not unknown in the US and Australia. In the US it is often included on Yiddish word lists, which is technically correct, although depending on context you may be more likely to hear Yiddish speakers use sha, sha shtil, or farshvaygn.