What is "Miss Maudie's goat" in "To Kill a Mockingbird"?

Question

From chapter 5 of To Kill a Mockingbird by Harper Lee:

We saw Uncle Jack every Christmas, and every Christmas he yelled across the street for Miss Maudie to come marry him. Miss Maudie would yell back, ’Call a little louder, Jack Finch, and they’ll hear you at the post office, I haven’t heard you yet!’ Jem and I thought this is a strange way to ask for a lady’s hand in marriage, but then Uncle Jack was rather strange. He said he trying to get Miss Maudie’s goat, that he had been trying unsuccessfully for forty years, that he was the last person in the world Miss Maudie would think about marrying but the first person she thought about teasing, and the best defence to her was spirited offence, all of which we understood clearly.

Please explain the lines in bold.

Also, why did Miss Maudie compare Uncle Jack’s playful nature with Jem’s ideas about Boo Radley ("He gets more like Jack Finch every day")?

Answer 1

Jack Finch is teasing Miss Maudie, playfully but unsuccessfully trying to annoy her.

To get somebody's goat, as Matt Thrower said in comments, means to make them annoyed or angry. As it says at the start of the paragraph, Miss Maudie and Jack Finch had known each other since they were children. He feels that she teases him a lot (he is "the first person she [thinks] about teasing"), and that the best way to respond to it is for him to tease her in return: "the best defence to her was spirited offence", meaning that he takes a proactive stance in teasing her as the best way to cope with her teasing him.

To say "he had been trying unsuccessfully for forty years" means that he's been trying to make her actually annoyed (to get her goat) since they were children together, but she is too unflappable. Even when he shouts inappropriate marriage proposals to her in the streets, she responds by teasing him right back.

As for why she compares Jem to Jack Finch, it seems that Jem's reported remark about Boo Radley displays a similar idea of humour. Because Boo Radley stays in his house and doesn't come out, Jem says he thinks that "he died and they stuffed him up the chimney". To come up with such a notion betrays the same sort of irreverent sense of humour that might lead someone to shout non-serious marriage proposals in the street.