This text tries to explain how digital signatures work, but it actually is a bit misguiding.
It explains digital signatures based on public-key encryption schemes, but this is only valid for schemes like RSA. And even here, you don't simply encrypt some message with your private key to sign it, but you apply the "decryption" operation (i.e. exponentiation with the private key) on a hash of a message, suitably padded.
In RSA, you have a key pair consisting of private key $(n, d)$ and public key $(n, e)$, where $n$, $d$ are quite big numbers (the size of $n$ relates to the security of the scheme), $e$ usually a smaller number like $2^{16}+1$.
In the "textbook version" (without any padding/hashing), to sign a message $M \leq n$, you calculate $s := M^d \bmod n$, where $\bmod$ is the integer remainder operator. The pair $(M, s)$ is then the signed message.
To verify the signature, anyone can calculate $M' := s^e \bmod n$ and then check if $M \mathrel{\overset?=} M'$. Here signing looks like RSA-encrypting with the private key and verifying looks like RSA-decrypting with the public key, which is the reason that the article tries to show it as such.
The real RSA-based signature schemes work a bit different: We apply a one-way hash function $H$ and a semi-random padding $P$ on the message before the exponentiation, so the signature is $s := P(H(M))^d \bmod n$, and this can be checked as $P^{-1}(s^e \bmod n) \mathrel{\overset?=} H(M)$ (where $P^{-1}$ also checks if the padding was really correct). This hashing and padding actually makes the scheme secure against forgeries (and it also allows to sign messages longer than $n$).
There are other signature schemes which do not relate directly to an encryption scheme, for example DSA.
To use a digital signature for authentication (or "identity verification", as the article says), you normally don't sign an arbitrary message, but you sign a piece of data which is somehow relevant to the protocol. For example, in the SSH protocol, at the end of a Diffie-Hellman key exchange, the server signs about everything important which was exchanged, to make sure that there was no man-in-the-middle attack which modified the exchanged data.
