The action potential causes Ca2+ ions to be released in the cells, that is needed for muscle fiber contraction. There is a small time delay from the arrival of the action-potential until the Ca-ions are released and reach the required concentration in the cell for contraction. On the molecular level there is a protein called troponin (this is bacically located on the actin filament along with tropomyosin and obscures the myosin binding site(s)), that bind calcium and then undergoes conformational change - this also adds up to the delay. After this - the conformational change of troponin - myosin and actin can bind together and contraction can occur through conformational changes. All these binding and conformation changes take a small time to be completed, and thus causes a small delay. Below is a picture about the molecular mechanism ofof muscle contraction taken from this wiki page. The linked wiki page has real good detailed information.
I found this article and where they exactly measured things. To clarify my answer (taken from the linked article):
Electromechanical delay (EMD) was described as a time elapse between the onset of muscle electrical activation and onset of force production, reflecting both electrochemical processes [i.e., synaptic transmission, propagation of the action potential, excitation-contraction (EC) coupling] and mechanical processes [i.e., force transmission along the active and passive parts of the series elastic components (SECs)]
So as for the whole muscle, not for just one cell the delay is caused by the things listed in the above quote - although excitation-contraction (EC) coupling and force transmission (things I have described earlier) take place in every single muscle cell that is in a specific muscle fiber.
Edit 2: I found a good and simple ppt/pdf about AP propagation and muscle contraction here.