Searle is not, as you might at first anticipate when you see that he's on the "computers can't be conscious" side of the argument, some sort of woo-woo psychic-spiritualism philosopher; he's actually really grounded and relatively materialistic, though he despises the latter term. (He thinks that you should not respond to the problems of dualism by choosing one or the other side of the duality to commit to, and argue the other into nonexistence -- but rather by saying "oh I guess it was stupid to draw a line here in the first place.") Even the earlier simplification I gave, he "thinks computers can't be conscious", he'd object to directly. He thinks that you're a computer, and he thinks that you're conscious. He just doesn't think that you're conscious by virtue of being the computer that you are. So that's who we're talking about here.
Now, Searle says: we know, or think we know, that the brain is conscious, and it's conscious due to some aspect of the dance of molecules that's going on under the hood. And we happen to have this model of dances, called computation, which is defined in terms of abstract symbol-shuffling: the great strength of computation is that the 0s and 1s can be voltages and electric currents (as in transistors) or magnetic alignments of spins (as in hard drive) or how a gear is turned (as in the difference engine) or whether a resistance is infinite or low (as in your keyboard). They're just abstract symbols.
At its most abstract the Turing test suggests that any human language can be boiled down to these abstract intrinsically-meaningless symbols (which is almost certainly true; that's the premise of the invention of writing), and then that anything which can output symbols in a way that's indistinguishable from a human being, deserves to be given the title of "understanding" that language.
Searle steps in and says, no, that's crazy. And he says it's crazy because he's a philosopher and philosophers are very concerned with what words mean exactly, and the word understanding means exactly that these things you're talking about are not abstract symbols, but a computer necessarily treats them that way; it's in the definition of computation that the symbols are abstract.
Now that you understand who this is and where he's coming from, maybe you can understand the argument better. He's trying to try to give a more formal proof of this above statement. So, he steps in with this Chinese room argument.
The Chinese Room argument says: Your brain is conscious, because of the dance that its neurons do. Now according to this nice computer-reductionalist view, anything which does any dance is understanding a language as long as it produces results which are indistinguishable from a human's responses. So Searle says: let's step inside some Turing machine's dance so our brains are doing two dances at once. The very asset which makes computers awesome allows us to take a Turing-test-passing-algorithm and perform it ourselves: we are the computer, we get this stream of symbols in, we look up rules in some complicated rulebooks, we stream some symbols out, and we now pass the Turing test for understanding Chinese. But is the test correct: do we actually understand Chinese? No. There is no reasonable definition by which we understand Chinese just by performing this complicated computational dance. And this applies even if we move all of the rulebooks and whatever into your skull so that the neurons do those things.
Now here's where we come to the conclusion that strong AI is impossible: the human has everything which is part of the definition of computation. I'll repeat that twice more. The human embodies a whole, complete, computer. There's nothing which a computer can be (by virtue of being a computer) which the human is not. So if we go this far and we say "that brain does not understand Chinese" we have to carry that over; "that silicon does not understand Chinese, either." And no computer will understand Chinese merely by virtue of running this computer program. Clearly our understanding of Chinese is not a property of arbitrary dances of neurons which can solve a Turing test, but there is something special about the right dance of neurons which solves a Turing test. The mere fact that the right dance exists points to some physics which is not reducible to computation.
In other words Searle thinks that you can simulate people as much as you please, but that simulations are not reality, thank you very much. Just like we humans are very good at simulating pain-behavior without actually being in pain, so too can a computer simulate understanding-behavior without actually understanding a thing. If you want to understand how understanding works, computers will doubtless be a useful tool, but they cannot offer a complete and final answer precisely because they do not pin down the material dance firmly enough to get to something causal.