Let's say we have two black boxes, named A and B. We want to see if they are the same system. The logical thing to do would be to feed both systems the same input data and see if they produce the same output. The more times we do this, and the more times we see the same outputs, the more certain we can be that they are the same system.

The Turing test, at a basic level, tries to take advantage of this fact. It indirectly compares two systems; a human and AI, to see if there are any distinguishable differences in output. It claims that if there are no observable differences in output between a human and AI, then we should regard the AI as sentient.

Let's return to systems A and B. Let's suppose they give the same outputs, given identical or similar inputs. Can we really be certain that the wiring inside these boxes is exactly the same?

Of course not. If we assume for a moment that the two black boxes take a series of numerical inputs and produce an output, based on a series of algorithmic calculations, we could have the following:

Algorithm A:
1) Take input x and add 10
2) Multiply x by 2
3) Add these results together

In order: output = x + 10 + 2x

Algorithm B:
1) Take input x and add 5
2) Multiply x by 4
3) Add these results
4) Subtract two times x
5) Add 5

In order: output = x + 5 + 4x - 2x + 5

Functionally, these algorithms calculate the same thing, only one of them in a more convoluted way than the other. In practice, this would make for two black boxes with different internal wiring, but we would have no way of telling from the outside. From an outsider's perspective, they are the same system and there is no way to distinguish them.

So this brings me to a theorem:
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Theorem 1: Given two black boxes, A and B, if inputs and corresponding outputs for both are the same, then either the internal wiring of A and B are the same, or one is a more efficient version of the other.
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We can state an alternative version of this theorem:

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Theorem 2: We have two black boxes, A and B. Instead of a single input-output system, we now have a range of outputs for a single input. For a single input, i, A can have outputs (X1, X2, ... XN) and B can have outputs (Y1, Y2, ... YN).

We can do a similar thing to theorem 1, and test the potential outputs of B for an input i. If we find that all outputs for B fall within the output range of A, (X1, X2, ... XN), for a particular i, then it is likely that black boxes A and B are the same type of system. (Of course we would have to test a wide range of i's to be certain).

This is simply a scaled up version of theorem 1, with multiple potential outputs.
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Now back to the Turing test. What we are basically doing, on a fundamental level, is comparing two systems; humans and AI. As humans we are all too familiar with the nature of human interactions, so we can safely compare them and a potential AI interaction.

We are seeing if the speech output of the AI falls within the range of acceptable human speech.

In theorem 2, if system A is a human and system B is an AI, then a human for a particular speech prompt, i, can have a range of speech outputs (X1, X2, ... XN). If for the same prompts the AI's (system B) output always falls within the range of acceptable human speech outputs, (X1, X2, ... XN), its passed the Turing test. Furthermore, by theorem 2, a human and AI can be regarded as the same type of systems.

In other words, the Turing test is testing theorem 2.

Therefore if an AI passes the Turing test (over a very large number of conversations), it is likely that the AI can be considered to be a "brain-like system", and therefore conscious.

I hope this all makes sense.