"Surely he hath borne our griefs,
and carried our sorrows:
yet we did esteem him stricken, smitten of God, and afflicted.
But he was wounded for our transgressions,
he was bruised for our iniquities:
the chastisement of our peace was upon him;
and with his stripes we are healed."
—Isaiah 53:4-5 (KJV)
The passage above is from the King James translation instead of the New Revised Standard Version that I use. The KJV is the one I found that uses the word 'stripes,' which is what you asked about. (Other versions translate the word as 'bruises'.)
One answer to your question would be that the word 'stripes' was the poetic choice of an unknown translator. The KJV was translated, not to be printed, but to be read aloud. Certainly 'stripes' is a more orally vivid word than 'bruises.'
This is from the fourth and final of the passages in Isaiah known as the Servant Songs. They suggest that the Messiah the people were seeking and demanding of God—someone to lead them once again out of bondage—would not be highborn, famous, and immediately obvious. He may be a servant, suffering in his mortality just as we do, and yet able to know and express the healing power of the Divine. It is often assumed that the reference is to the coming of Jesus some centuries later. That could be. What's certain is that he is describing for the Israelites a messianic leader who would emerge from the lowly, not from the aristocratic.
And so, on the metaphysical level, it is with us. We sometimes pray for massive, dramatic interventions in our lives, failing to notice the small, loving, and wonderful ways in which the Divine is already quietly at work within us.