Thirst is a universal sensation. Every human being gets thirsty, and I wouldn't be surprised if every human being ever born who was capable of it has actually said, "I am thirsty," out loud.
Why do we say this? It could be conversational, as we might announce, "I am tired," to whoever is within earshot. It could be explanatory, in case an observer might wonder why we're pouring ourselves a glass of water.
Many times, though, we aren't merely asserting a fact, we are revealing a need to someone whom we hope will help us.
That Jesus was thirsty on the cross shows He was human. That Jesus said He was thirsty shows He was needy.
If the thought of God making Himself needy doesn't make you tremble, then think of how we, in the persons of those with the galled wine, responded to His need.