I won't repeat what the other comments have said because they're correct about how evolution works. Although I will say it appears that mutations might not be as random as previously thought and then, of course, you throw epigenetics into the mix and it goes to show that we really still have a lot to learn about how evolution and its mechanics work.

My response to OPs question was probably a little bit anthropomorphic in nature but I found sometimes it's easier that way when communicating scientific stuff.

Different amino acid structure in the silk produces different types of silk with different properties. Sometimes the spider will manipulate the silk differently out of its spigots to suit its purpose. Spider silk is incredible stuff.

Spiders can have up to 7 different types of spider silk, for all sorts of different purposes. Each silk has a different protein structure or composition of different protein strands.

Some of that silk can be considered 'sticky' which is the stuff designed to catch prey. When walking around their web they will avoid the sticky stuff but when they do have to interact with the sticky silk they will use little claws on the end of their feed called tarsal claws which help them manipulate the silk.

You might be asking how they stick to the non-sticky silk or anything for that matter. This is interesting and relates to physics. They use Van der Waals forces which is a weak attraction between atoms. The hairs on the pads of the feet are exceptionally dense, they have hairs, upon hairs, upon hairs which give them millions of points of contact for each foot. The strength of attraction for each hair is exceptionally low but with all of them combined it gives the spider the grip to carry about 170 times its own body weight.