"Odds" are a way of stating a gambling bet.
Lets say you bet on a horse and the bookmaker offers "100:1 odds against" ("a hundred to one odds against.") That means that if you bet $10 and the horse loses, you lose your $10; if the horse wins, you keep your $10 and the bookmaker pays you $1,000. Such a bet is a "long shot." The bookmaker isn't going to give you 100:1 odds on a horse that has any real chance of winning. The odds are "a hundred to one against you." If, by some miracle, the horse wins, you've "beaten the odds."
The passage probably refers to depressing statistics about how often marriages end in divorce, and how often couples break up. The "odds are against" a relationship lasting. But, the writer says, if you follow his advice, you "stand a better chance of beating the odds," that is, your relationship is more likely to last.