Nerves from the brain command the muscles contract and relax. The muscles are attached to ligaments, and the ligaments are attached to bones. So when the muscles contract, the bones are moved. If you want more details, take a look at these videos.
Movement of the human body, called locomotion, requires the coordinated efforts of the motor branch of the nervous system, skeletal muscles, and the skeleton. Nerve impulses innervate skeletal muscles and the muscle contracts. This means the muscles shortens and thus pulls. Muscles are attached to bones by tendons and when they pull they pull bones moving them at joints.
Muscles can only pull; they cannot push. Therefore, they often work in antagonistic pairs, i.e. against each other. For example, if the biceps brachii in your upper arm contracts it pulls the forearm upwards, i.e. flexes it. This bends your forearm upwards. The triceps brachii, also in your upper arm, extends, i.e. straightens, your arm at the elbow joint by pulling the forearm back down.
Muscles allow movement. There are also nerves that are connected to the brain and receive and carry messages to and from it, including those that signal for movement. This is a very basic answer, sorry!