The video of Sean Carroll's Oxford-Cambridge lecture titled "God is not a good theory" is almost one hour long. I recommend taking the time to watch the entire video. However, for the benefit of people without an hour to spare I will summarize his argument for you.
He starts with a definition of theory as an idea about the universe that may be true or false. For almost all believers, god qualifies as a theory. However, god is not a precisely specified theory, and this is one of the substantial problems with the theory of god.
Concepts of god can be placed into at least three categories: Passive, Active, and Emergent. A passive god, as conceived by arm chair philosophers, is justified as fulfilling some requirement for making logical sense of our universe, such as the first cause, the unmoved mover, and a necessary being. This a passive conception because this god is not intervening to change any physical laws. An active conception of god is that of a creator and ruler who cares about human life, communicates to humans about proper human conduct, performs miracles, grounds morality, organizes an after life. The active god has an empirically observable presence and is justified accordingly. An emergent conception defines god as synonymous with love, the universe, the laws of nature, feelings of awe/transcendence. An emergent god is justified as serving a rhetorical function.
Sean Carroll dismisses the emergent conception of god as unworthy of further discussion because we can have the same conversations about the same topics without making any references to a rhetorical god. An emergent god therefore is superfluous.
The passive conception of god has a huge problem. It is based on a-priori metaphysics. It is rooted in rationalism rather than empiricism, it fails to give priority to observation. Such arm chair reasoning has never taught us anything factually true about the world. What it reveals, at its best, are consequences of axioms, and this can be useful, particularly in mathematics and logic. But it doesn't tell us which axioms are possibly true. Such a-priori reasoning cannot get us to the facts about what is actually true in our particular universe.
Sean Carroll then proceeds to argue that even if we take the arguments for a passive god more seriously than is merited by this major flaw in the underlying epistemology, they still do not succeed. God as a necessary being, first cause, and similar concepts are refuted by the fact that we can easily conceive of many possible, self-consistent, self-contained, coherent, eternal universes in the forms of various mathematical constructs with no god, no first cause, etc. Furthermore, at least one of these possible universes is plausibly our universe as it appears to represent a framework that correctly models our universe.
A counter-argument is that while it is possible to conceive of universes without god, those universe are infeasible because they lack a sufficient cause or explanation, they provide no answer to the "why" question. A legitimate universe explanation must answer the question why there is a universe and why it is this particular universe, therefore a god is required. Sean Carroll disagrees. You may prefer that there be an explanation for why this universe exists instead of another, or for why this universe exists instead of no universe, but our universe could just be. We associate causes with events because we experience our universe that way. Cause identification is linked to the overall context, so examining the same event from different perspectives will very often result in our reaching different conclusions about the cause. The context in which the universe appears is different from the context of our daily experiences. So analogizing from the contexts of our experiences within the universe to the context of the universe as a whole here is a weak analogy.
Sean Carroll states that he does not think that everything within the universe can be associated with a reason or a cause. Here is a short discussion of "Purpose and the Universe" with a video of Sean Carroll discussing the topic in more depth at an American Humanists Association meeting. He says "The universe itself doesn’t have a purpose, nor is there one inherent in the fundamental laws of physics. But teleology (movement toward a goal) can plausibly be a useful concept when we invent the best description of higher-level phenomena, and at the human level there are purposes we can create for ourselves."
The primary point here is that all such a-priori metaphysical arguments claims ultimately boil down to contingent empirical claims. Why must there be a sufficient reason for the universe? We are obligated to adopt a skeptical stance to such "must be", "necessary", types of assertions. It can then be argued that sufficient reason is needed because everything else has sufficient reason. But that is an empirical claim. Therefore, we must examine the god hypothesis like we examine all other hypothesis and look for the simplest coherent theory that explains the largest amount of data.
So does god give us a good theory on conventional scientific grounds? For a variety of reasons, the answer is no. Conservation of energy means there is no need for a first mover, chemistry means there is no need for a giver of life, natural selection means there is no need for a designer of the many different species of life. Neuroscience suggests that there is no need for a provider of consciousness and cosmology suggests there is no need for a creator. While these latter questions remain unsolved problems, there are multiple viable hypothesis and these questions appear to be resolvable using the same types of empirical methods that have successfully resolved the other questions without a god being needed.
Sean Carroll then identifies the Fine Tuning argument as the best empirical argument for God. He identifies several weaknesses to this argument. One is that we do not know what other possible universes would support life because we do not know enough about what different forms of life are possible and under what other conditions those different possible forms of life would be viable. Life may be possible in many other forms and as a result the phenomena of life may be much more generic and common to many different universes than the Fine Tuning argument assumes. What is needed for life is a very hard question to answer and we are not even close to knowing what percentage of possible universes would support some form of life. Another weakness of the Fine Tuning argument is that modern physics predicts a multiverse, and in a multiverse where the parameters vary we would expect to find ourselves in a region of the multiverse where the parameters appear to be finely tuned to support our existence.
Lastly, the question of the probability that god exists given the data is addressed. If we did not know anything about the actual universe, but we have this theory that there is a god who created the universe and who cares about us human beings, what would we expect the universe to be like? We know what the universe looks like so it can be tempting to say that god would make the universe exactly as we see it. But that is a biased approach. To tackle this question properly, we must try to start with a blank slate. And here we encounter a problem with the very low entropy of our universe during the Big Bang. It was about 10 to the -10 to the 120 smaller than its current value. Such extremely low entropy is incompatible with the existence of life, so if god created the universe to support life then we would predict our universe would have started with much higher entropy. This would have resulted in a universe with one galaxy instead of our universe with billions of galaxies that are unnecessary for life on earth.
There are other similar empirical arguments against the god hypothesis. The problem of evil, the problem of random suffering, and the problem of lack of clear divine instructions. No god ever told us that matter is made of atoms, the universe is billions of years old, people of different races, genders, etc. should be treated equally, and the like. Trying to salvage god by assigning to god the traits of elusiveness and vagueness is counter-productive since those are traits that evidence a weakness in the god hypothesis. We cannot have it both ways and say that god is evidenced by fine tuning but no other evidence can count against god. That is a double standard. God is much more ontologically problematic than a multiverse. God is an entirely different metaphysical category from everything else, ill-defined, unnecessary, whimsical, and frustratingly reclusive. We do better explaining the universe without the god hypothesis.