Lime is calcium carbonate. Which leads us to one easy answer for why it might help tomatoes - to combat blossom end rot. Blossom end rot is caused by a calcium deficiency in the plant cell walls. Without enough calcium the cells cannot form correctly and break down in the fruit production of the plant. How does lime correct this? Obviously since lime is partially made of calcium it does add some to the soil but it also adjusts the pH levels of the soil. Lime will raise the pH (per Hydrogen) of the soil and will make the soil less acidic. Plants have certain pH levels that are ideal for their growth and tomatoes prefer to be between 6.0 and 6.8 which is slightly on the acidic side. Neutral would be 7.0. Usually there is plenty of calcium in the soil for the plant to use for fruit growth but sometimes it isn't available for the plant because of various reasons like the wrong pH, too much water, or not enough water.
Having a good pH helps microbes break down materials better and improves the transfer of nutrients into usable forms for plants.
Before you go about adding lime to your soil it is a good idea to have a soil test done to check for the nutrient content and the soil acidity. You can buy tests to check the pH or you can utilize your local agricultural extension service. Most agricultural extension services will soil test for you at a very reasonable rate. They can give you some valuable feedback too. Our University of Tennessee extension can give recommendations based on the types of crops you want to grow.
As plants grow and break down over time the soil tends to gradually become more acidic. How much depends on a number of factors so it is best to get a test done on your garden soil every now and then to gauge what is happening in your soil.
How much lime should you add? If you've had a soil test you probably know exactly how much lime you need. If you aren't going to get a soil test then you can work a small amount into the top couple inches of soil before you plant (if you don't know the soil chemistry you may not even need it). Lime takes time to break down so it probably won't be immediately available for your plants. Adding lime in the fall after you've cleaned out the garden should give lime plenty of time to break down and make your soil a better environment for growing your tomatoes!