Commentary for Bava Kamma 3:5

This is all clear

היה בעל קורה ראשון וכו' פטור – for this one walks in an appropriate manner and the other one hurried to go.

This one comes carrying his jar and another one comes carrying his beam: this one’s jar is broken by that one’s beam, [The owner of the beam] is exempt, since this one has the right to walk along and this one has the right to walk along.
If the owner of the beam came first and the owner of the jar came after, and the jar was broken by the beam, the owner of the beam is exempt. If the owner of the beam stopped [walking suddenly], he is liable. If [the owner of the beam had said] “Stop” to the owner of the jar, he is exempt.
If the owner of the jar came first and owner of the beam came after, and the jar was broken by the beam, [the owner of the beam] is liable. If the owner of the jar stopped [walking suddenly],he is exempt. If [the owner of the jar had said} “Stop” to the owner of the beam, he is liable.
This mishnah continues to deal with the subject of damages done by a human being in the public domain. The basic topic is damages done by people carrying different objects and bumping into each other. One should note that although this mishnah (and most mishnayoth) deal with specific topics, their relevance is far greater than the immediate material learned. We are learning here about such important issues as individual rights which conflict with the individual rights of others. From the specific examples of the mishnah we can learn many principles about the public rights that people have, rights that may occasionally come into conflict with others’ rights. As one learns these mishnayoth please try to extract from them general principles and think about how they apply in other circumstances, perhaps circumstances relevant to modern societal problems.
This mishnah lists three scenarios regarding people walking in the public domain, one carrying a beam and one carrying a jar. In all of the scenarios the jar is broken by the beam and the mishnah needs to decide if the owner of the beam is liable. In section one the beam-owner and jar-owner are evidently walking together, meaning neither one is walking in front of the other. The owner of the jar has the right to walk in the public domain without being damaged. Therefore if the owner of the beam walks into him he is liable. He should have watched where he was going.
In the second clause the owner of the beam was walking in front of the owner of the jar. If the jar-owner walks into him, it is his own fault and the beam-owner is exempt. If, however, the beam-owner stops suddenly he is obligated. He should have realized that stopping suddenly might cause the other person to walk into him. Therefore if he didn’t stay “stop” he is liable, but if he did say “stop”, thereby warning off the jar-owner he is exempt.
The third section reverses the scenario that we saw in the second section: this time the jar-owner is walking first. Therefore if the beam-owner walks into him he is obligated. If, however, he stopped suddenly without warning the beam-owner the beam-owner will be exempt.