1. Disclaimer: Renormalization is a huge subject with many facets, such as, e.g. overlapping divergences of subgraphs, regularization, renormalization group, etc. Here we will only elaborate on OP's quote from Ref. 1.

2. Ref. 1 is considering a Feynman diagram ${\cal F}(q_1, \ldots, q_E)$ in momentum Fourier space, with external 4-momenta $(q_1, \ldots, q_E)$, and with internal 4-momenta $(p_1, \ldots, p_I)$, which are integrated over. The $p$-integrations are assumed to be UV divergent with positive superficial degree of divergence $D\geq 0$.

3. Here $E=\sum_f E_f$ is the total number of external lines, and $E_f$ is the number of external lines of field type $f$.

4. If we differentiate the Feynman diagram $D+1$ times wrt. the external 4-momenta, the integrand becomes UV finite. We conclude that the divergent part of the original Feynman diagram ${\cal F}(q_1, \ldots, q_E)$ is a polynomial in $(q_1, \ldots, q_E)$ of order $\leq D$. Note that the coefficients of the polynomial are possibly infinite!

5. We next add new interaction terms to the Lagrangian density ${\cal L}$ corresponding to $E$-vertices with $E_f$ fields of field type $f$, and possibly a finite number of spacetime derivatives (which in momentum Fourier space becomes a momentum monomial). The new interaction terms are so-called counterterms.

6. Feynman instructs us to sum over all Feynman diagrams with $E_f$ external legs of field type $f$. In particular, we should also include diagrams consisting of a single $E$-vertex coming from the new interaction counterterms. By adjusting possibly infinite coupling constants in front of the new interaction counterterms, the full Feynman diagram can be made finite.

References:

1. S. Weinberg, Quantum Theory of Fields, Vol. 1; Section 12.2, p. 506.

1. Disclaimer: Renormalization is a huge subject with many facets, such as, e.g. overlapping divergences of subgraphs, regularization, renormalization group, etc. Here we will only elaborate on OP's quote from Ref. 1.

2. Ref. 1 is considering a Feynman diagram ${\cal F}(q_1, \ldots, q_E)$ in momentum Fourier space, with external 4-momenta $(q_1, \ldots, q_E)$, and with internal 4-momenta $(p_1, \ldots, p_I)$, which are integrated over. The $p$-integrations are assumed to be UV divergent with positive superficial degree of divergence (SDOD) $D\geq 0$. Concerning SDOD, see e.g. my related Phys.SE answer here.

3. Here $E=\sum_f E_f$ is the total number of external lines, and $E_f$ is the number of external lines of field type $f$.

4. If we differentiate the Feynman diagram $D+1$ times wrt. the external 4-momenta, the integrand becomes UV finite. We conclude that the divergent part of the original Feynman diagram ${\cal F}(q_1, \ldots, q_E)$ is a polynomial in $(q_1, \ldots, q_E)$ of order $\leq D$. Note that the coefficients of the polynomial are possibly infinite!

5. We next add new interaction terms to the Lagrangian density ${\cal L}$ corresponding to $E$-vertices with $E_f$ fields of field type $f$, and possibly a finite number of spacetime derivatives (which in momentum Fourier space becomes a momentum monomial). The new interaction terms are so-called counterterms.

6. Feynman instructs us to sum over all Feynman diagrams with $E_f$ external legs of field type $f$. In particular, we should also include diagrams consisting of a single $E$-vertex coming from the new interaction counterterms. By adjusting possibly infinite coupling constants in front of the new interaction counterterms, the full Feynman diagram can be made finite.

References:

1. S. Weinberg, Quantum Theory of Fields, Vol. 1; Section 12.2, p. 506.

2. M.E. Peskin & D.V. Schroeder, An Intro to QFT, 1995; Section 10.1, p. 319.

1. Disclaimer: Renormalization is a huge subject with many caveats, such as, e.g. overlapping divergences of subgraphs, regularization, etc. Here we will only elaborate on OP's quote from Ref. 1.

2. Ref. 1 is considering a Feynman diagram ${\cal F}(q_1, \ldots, q_E)$ in momentum Fourier space of superficial degree of divergence $D\geq 0$, and with external 4-momenta $(q_1, \ldots, q_E)$.

3. Here $E=\sum_f E_f$ is the total number of external lines, and $E_f$ is the number of external lines of field type $f$.

4. If we differentiating the Feynman diagram $D+1$ times wrt. the external 4-momenta, the new Feynman diagram becomes UV finite. We conclude that the divergent part of the original Feynman diagram ${\cal F}(q_1, \ldots, q_E)$ is a polynomial in $(q_1, \ldots, q_E)$ of order $\leq D$. Note that the coefficients of the polynomial are possibly infinite!

5. We next add new interaction terms to the Lagrangian density ${\cal L}$ corresponding to $E$-vertices with $E_f$ legs of field type $f$ and possibly a finite number of spacetime derivatives (which in momentum Fourier space becomes a momentum monomial).

6.   The new interaction terms are so-called counterterms. Feynman instructs us to sum over all Feynman diagrams. In particular we should also include the new interaction terms. By adjusting possibly infinite coupling constants in front of the new interaction terms, the full Feynman diagram can be made finite.

References:

1. S. Weinberg, Quantum Theory of Fields, Vol. 1; Section 12.2, p. 506.

1. Disclaimer: Renormalization is a huge subject with many facets, such as, e.g. overlapping divergences of subgraphs, regularization, renormalization group, etc. Here we will only elaborate on OP's quote from Ref. 1.

2. Ref. 1 is considering a Feynman diagram ${\cal F}(q_1, \ldots, q_E)$ in momentum Fourier space, with external 4-momenta $(q_1, \ldots, q_E)$, and with internal 4-momenta $(p_1, \ldots, p_I)$, which are integrated over. The $p$-integrations are assumed to be UV divergent with positive superficial degree of divergence $D\geq 0$.

3. Here $E=\sum_f E_f$ is the total number of external lines, and $E_f$ is the number of external lines of field type $f$.

4. If we differentiate the Feynman diagram $D+1$ times wrt. the external 4-momenta, the integrand becomes UV finite. We conclude that the divergent part of the original Feynman diagram ${\cal F}(q_1, \ldots, q_E)$ is a polynomial in $(q_1, \ldots, q_E)$ of order $\leq D$. Note that the coefficients of the polynomial are possibly infinite!

5. We next add new interaction terms to the Lagrangian density ${\cal L}$ corresponding to $E$-vertices with $E_f$ fields of field type $f$, and possibly a finite number of spacetime derivatives (which in momentum Fourier space becomes a momentum monomial). The new interaction terms are so-called counterterms. 

6. Feynman instructs us to sum over all Feynman diagrams with $E_f$ external legs of field type $f$. In particular, we should also include diagrams consisting of a single $E$-vertex coming from the new interaction counterterms. By adjusting possibly infinite coupling constants in front of the new interaction counterterms, the full Feynman diagram can be made finite.

References:

1. S. Weinberg, Quantum Theory of Fields, Vol. 1; Section 12.2, p. 506.