The volume of the reaction vessel within a bomb calorimeter is fixed. This means that the heat flow between the reaction vessel and the surrounding water bath must equal the internal energy change within it. In other words, none of the change in internal energy goes into pressure-volume work.
Thermochemistry in General Chemistry is conceptually analogous to the discussion of the First Law of Thermodynamics in Physics. Thermochemistry is conservation of energy applied to chemistry. In both topics, work means the same thing, i.e. the product of the pressure and the change in volume of the system. Furthermore, heat flow is the other way that systems exchange energy with their surroundings, although thermochemistry pins down heat flow in a certain way as a change in a state function enthalpy. When you define things so that heat flow equals the change in enthalpy it allows heat flows to be summed over alternative pathways, which is invaluable to chemical reasoning. The formalism that makes this possible in chemistry is to assume that pressure is constant over the path. In that case, the change in the enthalpy, ie. the change in the internal energy plus the pressure volume product, does equal heat flow.
One important thing to consider when applying the 1st law to chemistry is that internal energy is a much richer idea in chemistry. In physics, we kept pretty much to the ideal gas whose internal energy is only the kinetic energy of the particles. In chemistry, though, the internal energy is much more complicated. It helps to understand that there really are two primary kinds of internal energy relevant to chemistry. These are the kinetic energy, which will have rotational and vibrational modes in addition to translational modes in most chemical substances, and the electrostatic potential energy associated with the arrangement of charged particles (the protons and electrons in a particular atomic, molecular, or intermolecular arrangement). Crucial to being able to conceptualize chemical change in the context of the First Law of Thermodynamics is a clear sense of the electrostatic potential energy changes in chemical substances as they impact internal energy change. Which bonds are broken? Which bonds are formed? To understand the internal energy changes associated with a chemical process you have to look at the rearrangement of charge in the new bonding or intermolecular state of the system.