It appears to beis using the zlib format followed by Base64 coding, and then preceding the resulting string with "1:". So to use it externally, you can strip the "1:", do Base64 decoding, and feed the result of that to a zlib decoder.
However what you get out may not be immediately useful. I compressed the result of D[x^x, {x,9}], like one of the examples in the Compress[] documentation, and then decompressed (successfully) with zlib. I got what appears to be some sort of internal encoding. E.g. "!boRf" 0xa0, 0, 0, 0, "s", 0x04, 0, 0, 0, "Plusf", 0x03, ... (where the numbers are unprintable bytes).
If you want something interoperable, then use "GZIP" or "BZIP2" in ImportString and ExportString. For example, using a 100,000,000 byte excerpt of English from Wikipedia:
ExportString[enwik8,"GZIP"]//StringLength 36548933
ExportString[enwik8,"BZIP2"]//StringLength 29008736
Then you will also get to control the encoding of the data into a string to be compressed. And you can decide whether or not to encode the compressed data into a printable form, or leave it as binary.