The answer given by @Nicolai is essentially correct (and I have upvoted it). However I feel that the question embodies certain mistaken assumptions that should be challenged so that naïve readers of the question are not misled in to accepting them. (I refer to these below as ‘problems’.) I also feel that illustrations — not included in the answer provided by @Nicolai — are needed in this respect.)
I have used a 2016 paper in Science by Gao and Wang as my authority, quoting from it and reproducing part of one of its figures for those who have not library access to this journal. The authors actually propound a ‘three-metal ion’ mechanism, rather than the ‘two-metal ion’ mechanism mentioned by @Nicolai. This, however does not affect the basic points I am making, although it serves to illustrate that the details of the enzyme mechanism are of a complexity that is unsuitable for teaching at an introductory level.
Here are two key frames from Fig. 1 of that paper:
“I'm assuming that DNA polymerase is taking advantage of the released phosphates from ATP (and the appropriate forms of GTP, CTP, TTP) to become activated?”
An enzyme is not ‘activated’ by the products of the reaction it catalyses. The binding of the substrate may result in a conformation change that results in more favourable state for catalysis — and which might be referred to as ‘activation’‡ in a discussion of that topic; but the use of that word can only confuse at the level at which the current question is asked. This is because one of the key concepts in enzyme catalysis is ‘activation energy’ — the energy required to achieve the transition state. Activation energy refers to the activation of the reactants — not the enzyme. To quote from the paper:
Enzymes increase the rate of chemical reactions, which is thought to occur by a reduction in the activation energy required to reach the transition state (Fig. 1A)
Chemical reactions which result in a decrease in (Gibbs) Free Energy (the ‘Energy’ axis in Fig. 1A) are thermodynamically favourable. They occur slowly because they proceed through a transition state of higher energy. The role of an enzyme catalyst is to lower the energy to get to that transition state (the activation energy). As the figure shows, this reaction is thermodynamically favourable. The released pyrophosphate has no function, except to be hydrolysed by pyrophosphatases to prevent the polymerization being reversed.
It is unclear what the arrow at the left of the figure in the question (from the α-phosphate of the dNTP to the primer OH) is meant to represent†. Normally one would assume that this would be an electron, however here this does not make sense. As @Nicolai says
“…reaction proceeds by a simple nucleophilic attack of the 3'OH of the primer on the α-phosphate of the incoming dNTP”
i.e. the arrow should be from the oxygen electrons of the OH to the nucleophilic phosphate. This is shown in Fig. 1B (where the nascent DNA chain is on the left and the dNTP on the right). The latter figure actually shows the transition state, which allows one to see in a general way how this could be of lower energy than for the uncatalysed non-enzymic reaction. It is stabilized by the two metal ions (Mg2+), which themselves are held in the appropriate position by acidic side-chains at the active site of the enzyme.
‡As @user1136 points out, the expression ‘enzyme activation’ is also used to describe action of positive allosteric effectors — small molecules that increase the activity of the active site by a structural change initiated by their binding at a site distal from the active site — the allosteric site. This is a further potential cause of confusion as it has no relevance here.
†In a comment, the poster writes “The arrow in the… picture… represents conceptual movement of molecules and not ‘real-world’ chemical interactions”. If this is what was intended (I am not familiar with the source), in my opinion the authors should have used a different style of arrow (e.g. a wider one, ➡︎ rather than ➛, and not drawn it from the phosphate to the OH. It betrays a lack of either knowledge or concern for chemical conventions, and is particularly misleading for the student who wishes to understand biology at a chemical level. I find it inexcusable in a publisher with a reputation like that of Benjamin.