As Our Lord hung on the Cross, he made the simple statement "I thirst." Like so much else in Sacred Scripture, this is meant to be polyvalent, that is to have many meanings. On the surface level, it means just what it says. After probably sixteen or more hours, since He left the Last Supper, Christ was thirsty. It is very doubtful that in the process of interrogating and beating Him, the Jews or the Romans offered a glass of water [although, the Romans could have done so just in order to prolong the torture of the Cross; they liked for the crucified to be around for a while]. Our Lord thirsted. His lips were dry, as was His throat. He had lost much fluid by the shedding of His Precious Blood, as sweat in the Garden of Gethsemane; during the "interrogation" by the Sanhedrin; during the scourging, the via Crucis and at last the crucifixion itself. In spite of this, He was no doubt perspiring quite a bit as well, since it was in the heat of the day that He was crucified.
As we look through this level of the meaning, it is important to know that it shows the humanity of Christ. His Sacred Humanity was not a veil or a ruse. Like the statement by Saint Matthew, He had fasted for forty days and forty nights, and He was hungry, this shows the reality of the Sacred Humanity that Our Lord assumed for our sake. He was not able magically to avoid hunger and thirst because of His Divinity.
On the deeper layer, peeling away at the onion, as it were, Jesus was athirst for souls. Mother Teresa of Calcutta saw it as her mission in life to quench the thirst of her Spouse for souls. In every convent of the Missionaries of Charity there is a Crucifix in the chapel with the words I thirst or Sitio near it, because Blessed Teresa saw this a crucial passage of Scripture for her spirituality.
The service she gave to so many in the slums of Calcutta was because she viewed them as being another Christ. She saw them as "Jesus, in one of His many distressing disguises." Taking seriously the scene of the Judgement in St Matthew's Gospel, she sought to give Christ food when He was hungry; give Him drink when thirsty, etc. She cared for Christ in the least of His brethren. All of this, of course is the quench His thirst for souls.
Christ thirsts for souls, not just for numbers, but on an individual level. Each and every person. He thirsts for your soul, and for mine. He thirsts for the soul of the person down the street; for the soul of the sinner who needs to repent. He thirsts for the soul of the person who has wandered far away from his Father's house that is the One, Holy, Catholic and Apostolic Church. He thirsts for the soul of those who have always remained faithful, that they may be further perfected in Him.
It is in coming to Him to be purified that we can then become icons of Our Lord. If we have responded to His grace, others should be able to see Him in us! Then, as we go about ministering to Him in others, they will be brought to Him. We will quench His thirst.