Reddit reviews The Prime Numbers and Their Distribution (Student Mathematical Library, Vol. 6) (Student Mathematical Library, V. 6)

There's a pretty good outline of the stuff here. The third really follows from the previous two, fairly directly. The real connection between this product and the Euler-Mascheroni constant is then basically the fact that Merten's Constant is equal to (Euler-Mascheroni) + (Extra term involving primes). This is the hard part of Merten's Theorem and many modern treatments of the theorem just show that such a constant exists, but don't relate it to the Euler-Mascheroni constant (see this). This article, however, does do Mertens some justice. Though, This book does the same and presents it nicely, using more modern considerations (but I can't link a whole book to you).