The general meaning of cutting cards is as Peter says. But what exactly does a "rich deck" mean, and why should it be getting harder to cut? You get a general sense that he's lost again, that he has a lot of bad luck now, and that he used to be luckier and more successful when he was young. But when you know a bit about blackjack it makes more sense, but it requires a lengthy explanation.

Why blackjack specifically? Because of "bust card": that shows which game it is. Blackjack is a card game in which each player is initially dealt two cards. The object is for the sum of the value of the cards to be as high as possible without exceeding 21, counting face cards (jack, queen, king) as 10 and ace as 1 or 11, according to the player's preference. So if you're dealt a 10 or face card and an ace, you have 21, which is a winning hand.

If you have less than 21, you can ask for another card to see if you can get closer to 21, or you can leave it alone, hoping that the dealer, whom you are trying to beat (and whose cards you can't see) has a lower total than you. If you draw another card and it brings your total over 21, you lose automatically; you have a "bust hand", and the card you drew, that made you lose, is called a "bust card". So being "dealt a bust card" means losing, having bad luck.

To increase your chances of winning, you need to be able to estimate the chances of being dealt higher or lower cards. And that will depend on the order of the cards in the deck. So players use some kind of card-counting technique to give them an idea of the distribution of cards in the deck and the likelihood of being dealt favourable cards.

This is where "rich deck" and "cut" come in. The deck or decks (usually several are used) are shuffled at the start and then cut: a player inserts a coloured card, called the "cut card", into the deck at a point of his or her choosing, and the cards are cut at that point (the cards below the cut card are placed on top). The dealer inserts the cut card again. When all the cards down to the cut card have been dealt, the deck is shuffled again, so the bottom part is not used (otherwise skilful card counters would know what cards are left). The cards are placed in a box called a shoe and dealt from there.

A rich deck means that the upper part of the deck, above the cut card, has a high proportion of high-value cards: 10s, face cards and aces. This is favourable to the player; it increases the chances of getting a winning hand.

A deck with a lot of high cards is easy to cut; the coloured cut card is easier to insert than with low cards. This is because the high-value cards get slightly bent as play progresses, because the dealer bends them to check whether he has a blackjack (the details are complex, but trust me; see pp. 141-142 of the book cited below for an explanation). Conversely, a deck with a preponderance of low cards is harder to cut: it's harder to slide the coloured cut card into it.

So then this passage is comparing the character's life to a blackjack game. In his youth the deck was rich; it had a lot of high cards, he had more luck, he got a lot of good hands. Now that deck is getting harder and harder to cut, because it contains fewer and fewer good cards. He more often gets dealt a bust card; he more often has bad luck and loses.

Sorry for such a long explanation, but it's a very complex metaphor. Here are some references for further information:

On card counting and a "rich deck":
"The basic premise of card counting is that a deck or shoe that is 'rich' in face cards and aces (IE: a high percentage of these cards) favors the player over the dealer."
http://www.blackjackcenter.info/blackjack4.html

On cutting in blackjack:
https://books.google.es/books?id=LKN12w4vDCIC&pg=PA333&lpg=P...

On the deck being easy or hard to cut and bent cards and on how cards get bent:
"In most decks there are certain places where it is easy to cut the deck" (pp. 334-335) and pp. 141-142.
https://books.google.es/books?id=DrJ7Kozxo4YC&pg=PA334&lpg=P...

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Note added at 8 hrs (2015-05-21 12:12:04 GMT)
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Perhaps this very specific interpretation will seem rather contrived to some people. But I think we need to realise that this comes from a book set in Las Vegas, where the language of gambling, and specifically blackjack, will be much more familiar than in other places. As I say, "bust card" is specifically an expression used in blackjack. To say that the deck is becoming harder to cut is a strange expression, but to a blackjack player is makes sense.

So you might paraphrase the essential meaning of the metaphor by saying something like: "the good cards fate had dealt him in his youth had become more and more infrequent."