Why might Roger Penrose argue that Godel's incompleteness theorem(s) suggest that consciousness is non-algorithmic? 
This past March, when I called Penrose in Oxford, he explained that his interest in consciousness goes back to his discovery of Gödel’s incompleteness theorem while he was a graduate student at Cambridge. Gödel’s theorem, you may recall, shows that certain claims in mathematics are true but cannot be proven. “This, to me, was an absolutely stunning revelation,” he said. “It told me that whatever is going on in our understanding is not computational.”
–"Roger Penrose On Why Consciousness Does Not Compute" (2017-05-04)

Why might Roger Penrose think/suggest that Godel's incompleteness theorem(s) show that consciousness is non-algorithmic?
Clearly in this case answers can be speculative... (soft question)
 A: I'll give a simplified version of Penrose's reasoning. 
In a sense this goes back to the "liar paradox" that has been known for thousands of years. Suppose I say "I'm lying"; then that should mean that I am telling the truth; but that would mean that I am lying. Statements that refer to themselves, or which refer to each other in circles, can create unresolvable contradictions. 
Analogous paradoxes can be constructed for mathematics and computation. 
You can have a computer program which predicts whether another computer program will eventually stop running, or run forever; and then it has a nemesis program which contains a copy of the prediction program and always does the opposite of what it predicts. The prediction program inherently cannot win. Either it makes no prediction, or it makes the wrong prediction. 
Godel did something similar for a theorem-proving program. He was able to encode how the program works in arithmetic, and then write down an equation which implies that "The theorem prover says this equation is false". This nemesis sentence of the theorem prover is called the Godel sentence. Either the theorem prover "has no opinion" about whether the Godel sentence is true or false, or it gets caught in contradiction. 
This is the incompleteness theorem. If the prover is always correct, it must avoid taking sides on Godel sentences, or else it will fall into contradiction. To remain consistent in its assertions, its power to deduce the truth must be incomplete. 
The Godel sentence is possible because ordinary computation can be reduced to arithmetic operations on zeroes and ones, so facts about what a computer can and cannot do, can be expressed in arithmetic. However, you could have a special computer which, in addition to the usual logic gates, has a magic component which correctly outputs the answer to problems like "does this program halt" or "is this Godel sentence true". Mathematically, the magic component computes a function - it takes an input and produces an output - but it is not a function that can be implemented using arithmetic operations. Such a function can be called an oracle function. 
Now consider the capacity of the human brain to reason about mathematics, under the assumption that the human brain follows laws of physics. The known laws of physics involve computable functions. One might then conclude that there must be Godel sentences for the human brain too, mathematical statements which, even if true, are beyond the power of human reasoning. 
Penrose chose the other option. Human beings can reason correctly about Godel sentences, therefore the human brain must be able to employ oracle functions, and so physics must contain processes which require oracle functions for their definition. His concrete proposal (elaborated with Hameroff) is that human cognition employs quantum entanglement in the brain, and that quantum dynamics (especially the collapse of the wavefunction) is determined by subtle quantum-gravitational effects governed by an oracle-function law. 
A: Penrose has long been an advocate of the non-computational view of consciousness, and he is famous in this regard for advocating the supposed quantum mechanical nature of consciousness. Of course, this has given cover to woo-hoo artists, i.e. Deepak Chopra. But Penrose tried to actually present an argument based on Godel's incompleteness theorems, as you stated, yet all of these arguments rest on faulty assumptions. 
This article covers the history of Penrose's argument well (he was not the original author of the idea, and his ideas about it changed over the years), and gives a good survey for the various reasons Penrose thinks consciousness is non-computational. Penrose's arguments can be posed in various ways, but essentially they rest on the idea that consciousness emerges from some kind of heuristic abstraction inexpiable by classical physics. Similar to how intuitionist mathematics views math as an art, a non-linear, non-computational process, Penrose suggests something similar is at work with consciousness.

Why Penrose think/suggest that Godels incompleteness theorem(s) show that consciousness is non-algorithmic?

In the link above, the author states,

This is the clearest and most succinct formulation of the argument I know of): (1) suppose that “my reasoning powers are captured by some formal system F,” and, given this assumption, “consider the class of statements I can know to be true.”  (2) Since I know that I am sound, F is sound, and so is F’, which is simply F plus the assumption (made in (1)) that I am F (incidentally, a sound formal system is one in which only valid arguments can be proven).  But then (3) “I know that G(F’) is true, where this is the Gödel sentence of the system F’” (ibid).  However, (4) Gödel’s first incompleteness theorem shows that F’ could not see that the Gödel sentence is true.  Further, we can infer that (5) I am F’ (since F’ is merely F plus the assumption made in (1) that I am F), and we can also infer that I can see the truth of the Gödel sentence (and therefore given that we are F’, F’ can see the truth of the Gödel sentence). That is, (6) we have reached a contradiction (F’ can both see the truth of the Gödel sentence and cannot see the truth of the Gödel sentence).  Therefore, (7) our initial assumption must be false, that is, F, or any formal system whatsoever, cannot capture my reasoning powers.

For a specific rebuke of the whole idea, Max Tegmark's paper here demonstrates that the hypothesis of quantum mechanical origin for consciousness suffers from the decoherence of the quantum state at body temperature. Thus, quantum mechanics alone, as we understand it, does not produce consciousness in humans.
