It's made out of protein. Proteins are composed of many repeating chemical elements in a long string with actual physical bonds connecting them. This makes them incredibly strong in tension, but very weak for compression (i.e. holding things up).

Spider silk is mostly made up of silk proteins, which are quite amazing.You can make a model of spider silk with a string and some magnets that have holes in the center.Tie the magnets a few inches apart along the string. Push the string together so that the magnets attract and make the string shorter. That's the spider silk before you stretch it. Then pull on the string until all the magnets pull apart and the string is long. That's the stretched spider silk. When you stop pulling the string, the magnets pull together and make the string short again, like spider silk when it's not stretched. The string represents a really long protein molecule, and the magnets represent parts of the protein molecule that attach to each other. That's part of the reason why spider silk is so strong. Scientists are still trying to figure out all the reasons.

Here is a link where you can see a video about why spider silk is so strong: