Tackling this step by step:
Is a synonym for random chance, i.e. events that we cannot deterministically control the outcome of. I'll use the example of flipping a coin here - we don't know the outcome of the coin flip beforehand.
[The] play of blind chance
Is a fancy way of saying "the behavior of random chance".
Continuing the example of the coin flip, either outcome is 50% likely to happen. On average, if you make a large number of coin flips, let's say 2 billion, you expect to roughly get 1 billion heads and 1 billion tails.
But as you start flipping your coin 2 billion times and tallying the outcome, you won't always get a fair division between the two outcomes. Sometimes you might have flipped a few heads in a row, or the majority of recent flips will have been tails, or ... The numbers shift and change as you progress down your tall of coin flips, but it's likely going to average out in the end.
However, if you did another experiment where you flip a different coin 2 billion times, the progression of heads versus tails will be different from the first experiment.
This could lead you to wrongly conclude that each coin is individually different, but this is not the case. Even if you had used the same coin for the second experiment, you wouldn't have seen the exact same progression of heads versus tails.
This is the "behavior" of random chance, it does not always behave consistently in small numbers even though it does tend to average out in the long run. Individual observations of random chance can look different but this does not mean that they is an actual observable difference.
[A] is not the mere [B]
Is a fancy way of saying "is more than" or "is not just". It's indicating that A is either better than B, or that is B plus some other things as well.
For example, "Bob is not a mere violinist" could mean:
- He is a virtuoso, much better than the average violinist. ("Bob is better than a violinist")
- He is a violinist, but he also plays other instruments ("Bob is a violinist plus some other things as well")
It's not always clear which of the two is being used, this is very contextual.
[A] is not at all the mere [B]
"at all" is only used to strengthen the confidence of the statement. It does not change the meaning if omitted.
the sum of mental and physical phenomena known by the conventional name “person” or “individual”
Is a fancy way of referring to what we call "sentience", i.e. an object that displays physical mobility and rational thought, something of which we can state that it is alive and shows some form of intelligence.
Combining all this, we can reduce the sentence down to:
[Sentience] [is more than] [how random chance behaves]
In other words, the sentence is trying to convey that it would be incorrect to state that sentience is pure random chance and nothing else.