Brun's constant $1.9021605831\ldots$ is the sum of the reciprocals of twin primes greater than $2$, that is: $$\frac{1}{3}+\frac{1}{5}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13}+\frac{1}{17}+\frac{1}{19}+\ldots$$
Note $\frac{1}{5}$ is included twice as it's in two pairs of twin primes, but that's not material to this question.
Do I understand correctly, that if this constant were transcendental, that would prove the twin prime conjecture?
Conversely, if there are infinitely many twin primes, this would not necessarily prove that Brun's constant is transcendental.
If we were to assume that there are finitely many twin primes and then use this fact to somehow calculate Brun's constant, might we find that under this supposition Brun's constant must be transcendental and thereby, by contradiction, know that there are infinitely many twin primes?
Look at the problem in the contrapositive: if there are only finitely many pairs of twin primes, then the sum you show has only finitely many terms, and a sum of finitely many rational numbers is rational. Back in the original direction, if the sum is transcendental, then in particular it is irrational, and in that case the sum can't have only a finite number of terms. So if Brun's constant is transcendental, then there are infinitely many pairs of twin primes.
But as you say, even if there are infinitely many pairs of twin primes, this (now infinite) series is not guaranteed to converge to a transcendental, or even irrational, number. There are lots of infinite series that converge to rational numbers.