The Bible Knowledge Commentary says:

The verb “to have compassion” (splanchnizomai) is used in the New Testament only by the Synoptic Gospel writers: five times in Matthew (9:36; 14:14; 15:32; 18:27; 20:34), four in Mark (1:41; 6:34; 8:2; 9:22), and three in Luke (7:13; 10:33; 15:20). Suggesting strong emotion, it means “to feel deep sympathy.”

“Strong emotion” connotes passion and sympathy. Some go even further, saying that it carries the idea of “feeling with” someone—that is, entering into their experience and sharing it with them. That is certainly how Jesus is described in Hebrews 4:15:

For we do not have a high priest who cannot sympathize with our weaknesses, but One who has been tested in all things as we are, yet without sin. (nasb)

Jesus personally experienced the things we experience. Diving into the gospels, we see Jesus facing temptation (luke 4:1–13), hunger (luke 4:2), thirst (john 19:28), fatigue (john 4:6), rejection (john 6:60–66), being misunderstood (john 2:20–21), and more. He experienced (was “tested in”) all of the things that form our experience.

The writer of Hebrews said this holistic human experience enabled Jesus to be a proper high priest for us. In Israel’s way of government there were three primary offices: Prophet, Priest, and King. The role of the King was, among other things, to arbitrate between people. The task of the prophet was to represent God to the people. The duty of the priest, however was to represent the people to God. Jesus’s fully human experience equipped Him not to merely represent us before the Father in an official way but in an understanding way. He can be our sympathetic high priest because He has felt what we feel and experienced what we experience.

He has felt what we feel and experienced what we experience.

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That is the key. Although His experience was never marked by sin, He nevertheless felt sin’s effects on His creatures. This commitment by Jesus to share in the human experience is at the heart of His compassion for us.

So, then, where do we start when considering the compassion of Jesus? Interestingly, to explore the compassion of Jesus, we must begin by seeing compassion in the heart of the Father whom Jesus came to reveal to us.