What does late mean in this sentence?
Late in the winter of my seventeenth year...
One of the meanings of "late" is "towards the end of a period". The period can be an amount of time, such as a day, a week, a month, a year, etc, or a season. Late in the winter means when winter is almost over, when it is nearly spring.
Late is opposite of early: Early in the morning, late in the evening.
Early in the winter: end of December. Late in the winter: beginning of March.