Can I make my board larger?

Yes, you can scale Kanban board to better suit your available screen size, using the magnifying glass +/- icons available in Settings - Board Background.

Board Settings are only available to board owners and account admins/owners.

You could also design the board to take up more space by using wider columns - so that more than one card will fit in each column, horizontally. To do this, go to board Settings - Board Editor - pencil icon under a column - Lane Width. The number given in Lane Width determines how many cards will align one next to another in that column.