How did Romanticism influence Jane Austen? Not much if at all. The author of Pride and Prejudice and Sense and Sensibility lived from 1775 to 1817. The Romantic Period of English literature is usually dated from 1798, with the publication of Wordsworth and Coleridge's Lyrical Ballads, to 1850. One biographer notes that
she didn't sell a novel until 1803, her first actual publication was in 1811, and all of the novels whose first drafts had been written before 1800 were revised by her after 1809 before they were published -- so that her most important period of literary activity was 1810-1817.
So Romanticism was too new a "movement" to have much of an influence on her writing. In a letter to her sister, Austen states that she has just finished reading a poem by Byron, so she was aware of the literature.
The term "Romantic" does not refer to love and romance, but to a ideal of looking to imagination as the greatest mental faculty; we not only see the world around us but we create it when we think back on it and imagine it. Nature was important to the Romantics as well. They were the first "back to nature" buffs and believed that the "noble savage" was the happiest of all people.