I've heard the phrase 参った and understood it to mean something like 'I/we lost' or 'knocked out'. How did this come from 参る, to visit or go by? I read in a dictionary that it's some phrase said by a the defeated person in a judo or kendo match but I don't fully understand.
(I failed to identify the authoritative article about this, so the following story is based largely on my speculation.)
You probably know that 参る is a humble form and means to visit someone with higher status than ourselves.
And I think your dictionary also said that 参る can specifically mean "to visit a shrine/temple/grave." (=参拝) The noun お参り always means "a visit to a shrine/temple/grave and offering prayers." See the result of Google Image Search for お参り, and you can understand that Japanese 参る means not only "going", but also "showing the highest respect" for Gods, or someone else.
This, I suppose, explains why 参った also came to mean "I lost." By saying 参った, you admits your opponent is stronger than you, and deserves your respect. (In judo and kendo, showing such respect is especially important.)
There's a blog post about this topic.
By the way, this お参り pose (putting hands together in front of you) is also a gesture of surrender, or begging for one's life.