This Wikipedia article documents this algorithm:
For example, if N = 84923, we notice (by starting at 292, the first number greater than √N and counting up) that $505^2$ mod 84923 is 256, the square of 16. So (505 − 16)(505 + 16) = 0 mod 84923.
Computing the greatest common divisor of 505 − 16 and N using Euclid's algorithm gives us 163, which is a factor of N.
The importance of 292 makes sense to me since it's the first whole number after the square of N:
ghci> sqrt 84923
291.415510911825
But, I'm lost on the next part:
that $505^2$ mod 84923 is 256, the square of 16.
Please explain from which this part comes.
In that introduction, the article is motivating Dixon's method by first describing a simpler variant that doesn't use factor bases, namely Fermat's difference of squares factorization algorithm. If you read the algorithm listed there, you will see that it does a brute force search, looking for values of $\,a\,$ such that $\,a^2-N\,$ is a square, starting at $\,a = \lceil \sqrt{n}\rceil.\,$ Various sieving improvement's on Fermat's method are describe later in the article.