The phrase "matrimonial game" is a way of expressing the idea that every action and every thought in the novel surrounds the idea and necessity of marriage. One example is that Mrs. Bennet certainly sees having her daughters well married as her primary objective in life, making marriage seem like a game to her.
Mrs. Bennet is particularly worried about having her daughters well married because she knows the Longbourn estate is entailed to their nearest male relation, Mr. Collins. In Austen's time, women certainly were allowed to inherit if their was no son to inherit. However, such an inheritance was best passed on when there was only one child in the household and that child was a woman, just like Anne De Bourgh will be inheriting Rosings Park. Problems arose when the household consisted of multiple daughters as well as no male heir. Legally, if the estate was passed on to the women, every woman would inherit, which would actually break up the property and diminish its value. To prevent such loss, estates could be entailed to the next male family member (Republic of Pemberley, "Entail and Inheritance"). Hence, since Mr. Collins will be inheriting Longbourn after Mr. Bennet's death, his daughters will be left in financial distress if they do not marry well. Therefore, to keep wealth in the Bennet family, Mrs. Bennet knows that at least one of her daughters must marry a wealthy man, and seeing to this becomes her primary object, making it seem like a game.
We see Austen relay Mrs. Bennet seeing her daughters married as her primary objective in life in the very first description Austen gives of Mrs. Bennet, as we see at the end of the first chapter, "The business of her life was to get her daughters married" (Ch. 1). We further see just how obsessed Mrs. Bennet is with marrying her daughters through all of her actions, such as in the opening conversation that takes place in the first chapter. Here, she begs her husband to visit the newly arriving Mr. Bingley because he is young and known to be rich. When Mr. Bennet asks why he should visit Bingley and how his arrival into the neighborhood should affect their daughters, her reply is, "How can you be so tiresome! You must know that I am thinking of his marrying one of them," showing us just how fixated on marriage Mrs. Bennet is (Ch. 1).