This is an ill-worded attempt at explaining RSA signatures. The basic operation - modular exponentiation - is the same for all RSA operations. So in that sense you can think of RSA signature verification as "decrypting" with the public key.
However, both the type of key as the message padding are an integral part of RSA operations. They are required to secure RSA operations. RSA padding is different for encryption and signature generation, even if both are informally called PKCS#1 v1.5 padding. The input parameters are generally also different; the signature generation requires the padding to be applied on a specific formatted hash value internally.
Public and private key operations differ in several ways. Both use modular exponentiation. However, private keys require additional protection to make sure that the secret components that make up the private key aren't leaked through side channel attacks. Private key also used a large exponent or multiple values that allow calculations using the Chinese Remainder Theorem (CRT). Saying that signature generation is encryption with the private key is therefore a very dangerous statement; internally they should be quite different operations.
I would urge you to simply read the RSA PKCS#1 v2.1 specifications in RFC 3447. They are quite readable, and they make a clear distinction between the various primitives in section 5.2:
The main mathematical operation in each primitive is
exponentiation, as in the encryption and decryption primitives of
Section 5.1. RSASP1 and RSAVP1 [EDIT: for signature generation / verification] are the same as RSADP and RSAEP [EDIT: for decryption / encryption, in that order]
except for the names of their input and output arguments; they are
distinguished as they are intended for different purposes.
This was different for PKCS#1 v1.5 and earlier versions of the PKCS#1 standard. The change in wording in PKCS#1 was very deliberate. Unfortunately the confusion remains when it comes to the OID that specifies an RSA public key, even one that is used for signature verification:
{iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) pkcs-1(1) rsaEncryption(1)}
This is a legacy of the older PKCS#1 version 1 standards, I'm afraid.
For more information, also see https://crypto.stackexchange.com/q/15997/1172.