Learn Turkish: Sessizlikte birdenbire haykırasım geldi. - In the silence, I suddenly got an urge to cry out.

It combines the verb root haykır- (to cry out) with the suffix -ası and the first-person possessive -m. With geldi (it came), it literally translates to "my urge-to-cry-out came." This is the standard way to express a sudden, involuntary inclination to do something.

Why use geldi instead of var for this urge? Could I say haykırasım var?

You could use var, but it changes the meaning. Haykırasım var means you have a lingering, present feeling of wanting to cry out. Because this sentence includes birdenbire (suddenly) and refers to an urge that hit you at a specific moment in the past, the past tense geldi is the natural choice.

How is the word sessizlikte built from its root?

The root is ses (sound or voice). Adding -siz (without) makes the adjective sessiz (silent). The suffix -lik turns it into the abstract noun sessizlik (silence). Finally, the locative case suffix -te (in/at) gives us "in the silence." It is a great reminder of how Turkish stacks suffixes to build complex ideas from simple roots.

What is the difference between haykırmak and bağırmak here?