Why Augmenting Paths Work

When you flip edges along an augmenting path, the path starts and ends at unmatched vertices. Since the path alternates, flipping adds one more matched edge than it removes. If the path has length $2k+1$ (odd), you had $k$ matched edges before and $k+1$ after.

This increases the matching size by exactly $1$. You repeat this until no augmenting paths exist, which means you have found a maximum matching.