The straight segment of a racetrack leading to the finish line.
2.
Hence: (fig.) The last or concluding part of any extended activity that has a definite conclusion; the final phase of any activity; as, the election was neck-and-neck in the homestretch.
... I will make haste to get down to business, as Mr. Makely would say, for I am really coming to something that you will think worth while. One morning, when we had made half the circle of the capitals, and were on the homestretch to the one where we had left our dear mother—for Aristides claims her, too—and I was letting that dull nether anxiety for her come to the top, though we had had the fullest telephonic talks with her every day, and knew she was well and happy, we came round the shoulder of a wooded cliff ... — Through the Eye of the Needle - A Romance • W. D. Howells