.mid and .midi are the same thing. They are midi files. As you say this stands for Musical Instrument Digital Interface. They do not contain audio, but instructions to a midi device (such as piano, or a synthiser) on how to play a tune. They are the modern equivalent of the rolls used in a player piano. (if you check the sizes, you will find that they are tiny in comparison to an audio file - even a compressed one such as an mp3).
To convert them to audio, you will need to play them back using a midi device (most computer soundcards include midi capabilities) and record the resultant output. If you do this on a computer, they won't necessarily sound exactly as they did when played on your piano, in the same way that a player piano roll will sound differently depending on the piano on which it is played.
I convert midis to audio using a piece of software called MidiMaestro. This has a facility called "Wavemaker" which allows you to play the midi file and simultaneously re-record the output as an audio file.
http://www.midimaestro.com/