To solve this problem, we need to identify which of the given expressions represents a factorization from left to right. Let's analyze each option step by step:

- Option A: $(x+3)(x-3) = x^2 - 9$

This expression starts with a product of two binomials and results in a difference of squares. This process is known as polynomial multiplication, not factorization. Therefore, option A does not meet the criteria for factorization from left to right.

- Option B: $2ab - 2ac = 2a(b - c)$

This expression starts with a difference of two terms and ends with a product of a monomial and a binomial, where the common factor $2a$ is factored out. This process is exactly what factorization entails, starting with a polynomial and expressing it as a product of its factors. Therefore, option B correctly represents factorization from left to right.

- Option C: $(m+1)^2 = m^2 + 2m + 1$

This expression starts with a binomial squared and results in a trinomial. This process is known as polynomial multiplication, similar to option A, and does not represent factorization from left to right. Therefore, option C does not meet the criteria.

- Option D: $n^2 + 2n + 1 = n(n + 2) + 1$

This expression starts with a trinomial and attempts to express it in terms of a product of polynomials plus an additional term. However, this does not correctly represent the polynomial as a product of several polynomials, and thus, it does not meet the definition of factorization from left to right. Therefore, option D is incorrect.

Given the analysis above, the only option that meets the definition of factorization from left to right is:

$$\boxed{B}$$