Origins
Something suitable follows you perfectly. The word traces back to Latin sequī, meaning 'to follow', through Old French suir and Anglo-French suiter — 'to match, to fit'. When something suits you, it falls into line with your needs.
The Latin sequī is one of the most productive roots in English. From it we get sequence (things following in order), consequence (what follows from an action), pursue (to follow after something), prosecute (to follow through legally), and second (the one that follows first).
Development
A suit of clothes was originally a matching set — garments that follow the same pattern. A legal suit follows a grievance through court. A hotel suite is a set of rooms that follow one from another.
The -able suffix arrived in the 16th century, turning the verb suit into an adjective. Something suitable is literally 'able to follow' — capable of matching the situation. The word carries an older worldview where fitness meant alignment, where the right choice was the one that fell naturally into sequence with everything around it.