These numbers are the normalized likelihoods that the results given in the 10 toss vector
are obtained from the current distributions the coin A (or respectively B).

I'll work out the first two rows for illustration:

The guessed Bernoulli parameter for type A is 0.6 and for type B is 0.5.
According to the binomial distribution formula,
the unnormalized likelihood for obtaining 5H 5T are
From A:

L_A = C(10,5)*(0.6)^5*(0.4)^5

where C(10,5) is the binomial coefficient 10!/5!5!

Similarly from B we obtain:

L_B = C(10,5)*(0.5)^5*(0.5)^5

The normalized likelihoods are obtained as

For A: L_A/(L_A+L_B) = 0.4491

For B: L_B/(L_A+L_B) = 0.5509

For the second case 9H 1T

L_A = C(10,9)*(0.6)^9*(0.4)^1

L_B = C(10,9)*(0.5)^9*(0.5)^9

The normalized likelihoods:

For A: L_A/(L_A+L_B) = 0.8050

For B: L_B/(L_A+L_B) = 0.1950