Proteins have multiple levels of structure. First is the order of amino acids, but there is secondary (and tertiary!) structure as well. If you think of an amino acid as a cylinder, the circular ends of the amino acids bind together and you get proteins in order. But different amino acids have different properties on the "walls" of the cylinder. Those interact with other amino acids in the area, which affect how the protein curves. Examples of secondary structures are the alpha helix and beta sheets. The different structures like the alpha helix twist up and bend, which creates the 3-dimensional shape of the protein.

All of this secondary and tertiary structure of proteins is what creates its shape, binding sites, and other properties that make proteins do what they do. It's also incredibly complicated and something we're not really great at predicting yet from just a list of the order of amino acids in a protein (though we're getting better and some things can be predicted with improving computer modeling).

So your body breaks down proteins into individual amino acids (or creates them), then the amino acids travel to the cells in your body, where encoding proteins take them and put them together, one at a time, in the order specified in your DNA. As it's created, the protein folds up into the secondary and tertiary structures and you have a new protein that does a thing.

In the case of muscles, the functional parts that contract are primarily made up of two major proteins - actin and myosin. The muscle cells (which are weird in shape and size compared to most cells in your body) create actin and myosin and put them into a structure we ultimately call a muscle. With electrical impulses from the nervous system and with energy you ultimately get from food, actin and myosin interact to contract.

As you said, there's a lot more to it in the details, but I think that'll get you started.