In the PNG spec, uncompressed blocks include two pieces of header information:
LENis the number of data bytes in the block.NLENis the one's complement ofLEN
Why would the file include the one's complement of a value? How would this be used and/or for what purpose?
Rather than inventing a new compression type for PNG, its authors decided to use an existing industry standard: zlib.
The link you provide does not point to the official PNG specifications at http://www.w3.org/TR/PNG/ but only to this part: the DEFLATE compression scheme. NLEN is not mentioned in the official specs; it only says the default compression is done according to zlib (https://www.rfc-editor.org/rfc/rfc1950), and therefore DEFLATE (https://www.rfc-editor.org/rfc/rfc1951).
As to "why": zlib precedes current day high-speed internet connections, and at the time it was invented, private internet communication was still done using audio line modems. Only few institutions could afford dedicated landlines for just data; the rest of the world was connected via dial-up. Due to this, data transmission was highly susceptible to corruption. For simple text documents, a corrupted file might still be usable, but in compressed data literally every single bit counts.
Apart from straight-on data corruption, a dumb (or badly configured) transmission program might try to interpret certain bytes, for instance changing Carriage Return (0x0D) into Newline (0x0A), which was a common option at the time. "One's complement" is the inversion of every single bit for 0 to 1 and the reverse. If either LEN or NLEN happened to be garbled or changed by the transmission software, then its one's complement would not match anymore.
Effectively, the presence of both LEN and NLEN doubles the level of protection against transmission errors: if they do not match, there is an error. It adds another layer of error checking over zlib's ADLER32, and PNGs own per-block checksum.