Andy draws the question card, “Do you see two or more players whose cards sum to the same value?” He answers, “`yes.”

Sum of Belle's cards = 3 + 4 + 7 = 14
Sum of Carol's cards = 4 + 6 + 8 = 18

Since these have different sums, but Andy sees at least two players whose cards have the same sum, then your cards must add up to either 14 or 18

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Next Belle draws the question card, “Of the five odd numbers, how many different
Odd numbers do you see?” She answers “All of them.”

The only odd cards that Belle sees from Andy and Carol are 1, 3, 7

So you must have 5 and 9

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5 + 9 = 14

You know that sum of your cards = 14 or 18
Since 5 + 9 = 14, and smallest card is 1, then your cards must add to more than 14
Sum of your cards = 18

Third card = 18 - 9 - 5 = 4

Your cards: 4, 5, 9

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You can also see why Andy knew what cards he had. He realized that the only odd numbers Belle could see from Carol and you were 5 and 9, and yet she claimed she could see all five odd numbers. So the remaining three: 1, 3, 7 must have come from Andy himself. That's how he figured out what he had.
The logic is "Process of elimination", with more possiblities being eliminated every time new info comes up.
After A starts, you know you have a total of 14 (B) or 18 (C).
When B sees your 5 & 9, that takes out 14, your other card is