They're not creating energy, they're basically releasing energy stored over from the Big Bang. Triggering that release with less input energy than output energy is an engineering problem.
Consume a small amount of mass, release a very large amount of energy due to the huge conversion factor. The first law isn't violated because the energy being released is coming from the annihilation of matter.
Mass and energy can in principle be converted given the right process. LLNL consumed the mass that produced the energy.
In this specific case, hydrogen atoms are fused together, which is a reaction that releases energy because the resulting combined atomic nucleus is at a lower energy state than the atoms that went in. This technically holds true for fusion up until you get iron nuclei, with heavier atoms than iron requiring energy to be put into the reaction when fusing. This is the other side of the equation, which we are already using for nuclear fission plants.
Bashing nuclei together hard enough to overcome their innate repulsion is where most of the input energy is going in fusion reactors. It is comparable to the concept of activation energy in chemical reactions.
Keep in mind that this announcement does not cover the total energy cost of running the reactor, "only" the theoretical amount of energy released into the chamber by the ignition laser. So we're a long, long way off from achieving what they call "unity" (an input/output ratio of 1:1) in practical terms. In theory, fusion reactions achieving practical unity can be self-sustaining. Although this specific way of triggering a fusion reaction is not continuous, it could theoretically be re-worked into a permanently running reactor that is pulse-driven in the same way an internal combustion engine is.
At this point it sounds like the actual energy they put in is less than they get out, but the energy it takes to put the energy in still makes it a loss overall.
For every 1 deuterium nucleus involved, it fuses with 1 tritium nucleus => producing 1 helium nucleus, 1 free neutron, and 17.6 MeV of energy.
The energy released then causes more deuterium and tritium to fuse, producing more energy, and so on, in a chain reaction.
The goal, in order for this to be used to produce electricity for consumption, is for us to be able to kick start this process by introducing less than 17.6 MeV of energy per deuterium-tritium pair, so that after you account for the energy spent to fuse those atoms, there's a net energy left over for us to use.
Put even more simply, we're throwing heat/pressure at atoms, to cause them to fuse, and in the process convert some of that mass to energy.
Obviously causing a nuclear explosion is not a very good way to generate energy. :)